Q: In the equation $7x^2+2x-5=0$, what is the value of $a$?

7 In the standard form $ax^2+bx+c=0$, $a$ is the coefficient of $x^2$. In the given equation the coefficient of $x^2$ is 7, so $a=7$. Option B (2) is the coefficient of $x$ (that's $b$), and option C (−5) is the constant term $c$. Exam tip: Always write the quadratic in standard form $ax^2+bx+c=0$ and read off $a,b,c$ directly.

Q: What is the value of b in the quadratic equation $5x^2-9x+4=0$?

-9 A quadratic is written as $ax^2+bx+c=0$, where $b$ is the coefficient of $x$. In $5x^2-9x+4=0$ the coefficient of $x$ is $-9$, so $b=-9$. The option 5 is the coefficient $a$, 4 is the constant term $c$, and 9 is just the positive opposite of the correct value. Exam tip: rewrite the equation in the form $ax^2+bx+c$ and read off the coefficient of $x$ to avoid sign mistakes.

Q: In the equation $8x^2+3x+12=0$, what is the value of $c$?

12 A quadratic equation is written in standard form as $ax^2+bx+c=0$. Comparing $8x^2+3x+12=0$ with this form gives $a=8$, $b=3$ and $c=12$. Hence $c=12$. Option 8 is incorrect because it is the coefficient of $x^2$ ($a$), option 3 is the coefficient of $x$ ($b$), and 0 is incorrect because the constant term here is 12, not 0. Exam tip: rewrite any given equation in standard form first and then read off $a, b, c$.

Q: In the quadratic equation $11x^2-6x=0$, what is the value of the constant term $c$?

0 A quadratic is written as $ax^2+bx+c=0$. The given equation can be written as $11x^2-6x+0=0$, so the constant term is $c=0$. Note that $-6$ is the coefficient $b$ and $11$ is $a$; option D has the wrong sign. Exam tip: always put the equation in standard form and if the constant term is missing treat it as 0.

Q: In the equation $13x^2-20=0$, what is the value of $b$ in the standard quadratic form $ax^2+bx+c=0$?

0 Write the equation in standard form as $13x^2+0x-20=0$. Thus $a=13,\; b=0,\; c=-20$, so $b=0$. A common distractor is $-20$, but that is the constant term $c$, not the linear coefficient. Exam tip: if the x-term is missing, take the linear coefficient $b$ equal to zero immediately.

Q: Under which condition will the equation $px^2+qx+r=0$ be called a quadratic equation?

$p \neq 0$ By definition a quadratic equation must have the highest power of the variable equal to 2. That requires the coefficient of $x^2$, namely $p$, to be nonzero, i.e. $p\neq0$. If $p=0$ the $x^2$ term disappears and the equation becomes linear (or lower degree), so option A is incorrect. Options C and D ($q=0$ or $r=0$) merely eliminate other terms but do not change the degree as long as $p\neq0$. Exam tip: always check the leading coefficient to determine if an equation is quadratic.

Q: What is the standard form of $4x^2+7x=9$?

$4x^2+7x-9=0$ A quadratic is written in standard form as $ax^2+bx+c=0$. Moving the RHS term $9$ to the left changes its sign, giving $4x^2+7x-9=0$, so option B is correct. Option A is wrong because it keeps $+9$ instead of $-9$. Options C and D are wrong because they change the sign of the linear term $7x$. Exam tip: bring every term to one side and combine like terms to reach $ax^2+bx+c=0$.

Q: What is the standard form of the equation $2x^2=5x+3$?

$2x^2-5x-3=0$ The standard form for a quadratic is $ax^2+bx+c=0$. Move the right-hand side terms to the left by subtracting $5x+3$ from both sides to get $2x^2-5x-3=0$. Thus $a=2, b=-5, c=-3$. Option C is the closest distractor but has the constant term with the wrong sign (+3 instead of −3). Exam tip: always convert to $ax^2+bx+c=0$ by bringing all terms to one side before identifying coefficients or solving.

Q: If $x=3$, what is the left-hand side of $x^2-4x+3=0$?

0 Substitute $x=3$: $3^2-4\cdot3+3 = 9-12+3 = 0$. Thus the left-hand side equals $0$, so $x=3$ makes the expression zero. The closest distractor $3$ typically arises from sign or squaring mistakes (e.g. treating $3^2$ as $3$). Exam tip: evaluate each term separately and follow the order of operations (BODMAS).

Q: Which of the following is a solution of the equation $x^2-25=0$?

5 Solving $x^2-25=0$ gives $x^2=25$, so $x=\pm5$. The value 5 is among the choices and indeed satisfies $5^2-25=0$, so 5 is a correct solution. Checking the others: $2^2-25=-21$, $(-2)^2-25=-21$, $10^2-25=75$ — none are zero, so they are not solutions. Exam tip: take square roots to get $x=\pm\sqrt{25}$ and then substitute the available options to verify quickly.

Question 1

In the equation $7x^2+2x-5=0$, what is the value of $a$?

Accepted Answer

7 In the standard form $ax^2+bx+c=0$, $a$ is the coefficient of $x^2$. In the given equation the coefficient of $x^2$ is 7, so $a=7$. Option B (2) is the coefficient of $x$ (that's $b$), and option C (−5) is the constant term $c$. Exam tip: Always write the quadratic in standard form $ax^2+bx+c=0$ and read off $a,b,c$ directly.

Question 2

What is the value of b in the quadratic equation $5x^2-9x+4=0$?

Accepted Answer

-9 A quadratic is written as $ax^2+bx+c=0$, where $b$ is the coefficient of $x$. In $5x^2-9x+4=0$ the coefficient of $x$ is $-9$, so $b=-9$. The option 5 is the coefficient $a$, 4 is the constant term $c$, and 9 is just the positive opposite of the correct value. Exam tip: rewrite the equation in the form $ax^2+bx+c$ and read off the coefficient of $x$ to avoid sign mistakes.

Question 3

In the equation $8x^2+3x+12=0$, what is the value of $c$?

Accepted Answer

12 A quadratic equation is written in standard form as $ax^2+bx+c=0$. Comparing $8x^2+3x+12=0$ with this form gives $a=8$, $b=3$ and $c=12$. Hence $c=12$. Option 8 is incorrect because it is the coefficient of $x^2$ ($a$), option 3 is the coefficient of $x$ ($b$), and 0 is incorrect because the constant term here is 12, not 0. Exam tip: rewrite any given equation in standard form first and then read off $a, b, c$.

Question 4

In the quadratic equation $11x^2-6x=0$, what is the value of the constant term $c$?

Accepted Answer

0 A quadratic is written as $ax^2+bx+c=0$. The given equation can be written as $11x^2-6x+0=0$, so the constant term is $c=0$. Note that $-6$ is the coefficient $b$ and $11$ is $a$; option D has the wrong sign. Exam tip: always put the equation in standard form and if the constant term is missing treat it as 0.

Question 5

In the equation $13x^2-20=0$, what is the value of $b$ in the standard quadratic form $ax^2+bx+c=0$?

Accepted Answer

0 Write the equation in standard form as $13x^2+0x-20=0$. Thus $a=13,\; b=0,\; c=-20$, so $b=0$. A common distractor is $-20$, but that is the constant term $c$, not the linear coefficient. Exam tip: if the x-term is missing, take the linear coefficient $b$ equal to zero immediately.

Question 6

Under which condition will the equation $px^2+qx+r=0$ be called a quadratic equation?

Accepted Answer

$p \neq 0$ By definition a quadratic equation must have the highest power of the variable equal to 2. That requires the coefficient of $x^2$, namely $p$, to be nonzero, i.e. $p\neq0$. If $p=0$ the $x^2$ term disappears and the equation becomes linear (or lower degree), so option A is incorrect. Options C and D ($q=0$ or $r=0$) merely eliminate other terms but do not change the degree as long as $p\neq0$. Exam tip: always check the leading coefficient to determine if an equation is quadratic.

Question 7

What is the standard form of $4x^2+7x=9$?

Accepted Answer

$4x^2+7x-9=0$ A quadratic is written in standard form as $ax^2+bx+c=0$. Moving the RHS term $9$ to the left changes its sign, giving $4x^2+7x-9=0$, so option B is correct. Option A is wrong because it keeps $+9$ instead of $-9$. Options C and D are wrong because they change the sign of the linear term $7x$. Exam tip: bring every term to one side and combine like terms to reach $ax^2+bx+c=0$.

Question 8

What is the standard form of the equation $2x^2=5x+3$?

Accepted Answer

$2x^2-5x-3=0$ The standard form for a quadratic is $ax^2+bx+c=0$. Move the right-hand side terms to the left by subtracting $5x+3$ from both sides to get $2x^2-5x-3=0$. Thus $a=2, b=-5, c=-3$. Option C is the closest distractor but has the constant term with the wrong sign (+3 instead of −3). Exam tip: always convert to $ax^2+bx+c=0$ by bringing all terms to one side before identifying coefficients or solving.

Question 9

If $x=3$, what is the left-hand side of $x^2-4x+3=0$?

Accepted Answer

0 Substitute $x=3$: $3^2-4\cdot3+3 = 9-12+3 = 0$. Thus the left-hand side equals $0$, so $x=3$ makes the expression zero. The closest distractor $3$ typically arises from sign or squaring mistakes (e.g. treating $3^2$ as $3$). Exam tip: evaluate each term separately and follow the order of operations (BODMAS).

Question 10

Which of the following is a solution of the equation $x^2-25=0$?

Accepted Answer

5 Solving $x^2-25=0$ gives $x^2=25$, so $x=\pm5$. The value 5 is among the choices and indeed satisfies $5^2-25=0$, so 5 is a correct solution. Checking the others: $2^2-25=-21$, $(-2)^2-25=-21$, $10^2-25=75$ — none are zero, so they are not solutions. Exam tip: take square roots to get $x=\pm\sqrt{25}$ and then substitute the available options to verify quickly.

Question 11

What equation do we obtain when we substitute $x=4$ into $x^2-8x+16=0$?

Accepted Answer

0=0 Substituting $x=4$ gives the left-hand side $4^2-8\cdot4+16=16-32+16=0$, so the equation becomes $0=0$; therefore $x=4$ satisfies the equation. Option B (4=0) is a tempting distractor but incorrect because the left-hand side evaluates to 0, not 4. Exam tip: when checking roots, simplify squares and products first, then perform addition/subtraction to avoid arithmetic mistakes.

Question 12

For the equation 12x^2 = 0, what are a, b and c respectively?

Accepted Answer

(12, 0, 0) The general form of a quadratic is $ax^2+bx+c=0$. The equation $12x^2=0$ can be written as $12x^2+0x+0=0$, so the coefficients are $a=12,\ b=0,\ c=0$. The distractor (12,1,0) is wrong because the coefficient of x is 0, not 1. Exam tip: always write missing terms as $0x$ or $+0$ to read off coefficients correctly.

Question 13

Write $2x(x+7)=0$ in standard form $ax^2+bx+c=0$.

Accepted Answer

$2x^2+14x=0$ Core idea: use the distributive property. Multiply $2x$ across the bracket: $2x\cdot x=2x^2$ and $2x\cdot7=14x$, giving $2x^2+14x=0$. The closest distractor $2x^2-14x=0$ is wrong because the sign of the linear term is incorrect. Option $14x^2+2x=0$ swaps coefficients and is not equivalent; $2x+7=0$ is linear, not quadratic. Exam tip: always expand and collect like terms to write the equation as $ax^2+bx+c=0$ before choosing the answer.

Question 14

What is the expanded (standard quadratic) form of $(x+4)(x+5)=0$?

Accepted Answer

$x^2+9x+20=0$ Multiply each term: $(x+4)(x+5)=x\cdot x + x\cdot5 + 4\cdot x + 4\cdot5 = x^2+5x+4x+20$. Combining like terms gives $x^2+9x+20=0$, so option A is correct. Distractors fail due to sign or coefficient errors (D has the constant as $-20$; B and C have incorrect coefficients/signs). Exam tip: use FOIL (first, outer, inner, last) or multiply term-by-term and then combine like terms to avoid sign mistakes.

Question 15

What is the standard form of the equation $(x-6)(x+2)=0$?

Accepted Answer

$x^2-4x-12=0$ Multiply the factors: $(x-6)(x+2)=x^2+2x-6x-12=x^2-4x-12$. Thus the standard form is $(x^2-4x-12=0)$. Option B has the wrong sign on the x-term (+4x instead of -4x); option C shows -8x which results from incorrect addition of like terms. Exam tip: Use FOIL or multiply term-by-term and then combine like terms carefully.

Question 16

What is the coefficient of (x^2) in (x^2+12x+36=0)?

Accepted Answer

(1) A coefficient is the number multiplied by a variable term. In the expression $x^2+12x+36=0$, the squared term is written as $x^2$. When a variable or its power appears without a visible number before it, the hidden multiplier is 1. Thus, $x^2$ really means $1x^2$, so its coefficient is 1. The coefficient is not 12, because 12 belongs to the $x$ term, and it is not 36, because 36 is the constant term.

Therefore, option B is correct. A useful way to identify coefficients is to separate the terms: $x^2$ has coefficient 1, $12x$ has coefficient 12, and $36$ has coefficient 36 as a constant. The equation sign and the value on the other side do not change the coefficient of the squared term. Recognizing an unwritten coefficient of 1 is an important basic algebra skill.

Question 17

In the equation $ -2x^{2}+5x-1=0 $, what is the value of $a$ according to the standard form $ax^{2}+bx+c=0$?

Accepted Answer

-2 In the standard form $ax^{2}+bx+c=0$, $a$ is the coefficient of $x^{2}$. In $ -2x^{2}+5x-1=0 $ the coefficient of $x^{2}$ is $-2$, so $a=-2$. Option A (2) is a common trap that ignores the negative sign. Exam tip: write the equation in the form $ax^{2}+bx+c=0$ first and read off the coefficients including their signs.

Question 18

Which of the following expresses $x^2+10x+25=0$ in factorised (product) form?

Accepted Answer

$(x+5)^2=0$ $x^2+10x+25$ is a perfect square trinomial of the form $(a+b)^2=a^2+2ab+b^2$. Taking $a=x$ and $b=5$ gives $(x+5)^2=x^2+2\cdot x\cdot5+5^2=x^2+10x+25$, so the factorised form is $(x+5)^2=0$. The closest distractor $(x-5)^2$ is incorrect because $(x-5)^2=x^2-10x+25$, which has $-10x$ instead of $+10x$. Exam tip: compare the middle term with $2ab$ and check the sign to identify the correct binomial square.

Question 19

Which of the following is a solution of the quadratic equation $x^2-3x-10=0$?

Accepted Answer

5 Factor the quadratic: $x^2-3x-10=(x-5)(x+2)$. Thus the roots are $x=5$ and $x=-2$. Checking $5$: $5^2-3\cdot5-10=25-15-10=0$, so $x=5$ is a solution. A common distractor is $2$; checking shows $2^2-3\cdot2-10=4-6-10=-12$, not zero, so it is not a root. Exam tip: try factoring first; if factoring is hard use substitution or the quadratic formula to verify candidates quickly.

Question 20

If the highest power (degree) of the variable in an equation is $2$, what is that equation called?

Accepted Answer

द्विघात समीकरण (Quadratic equation) Core concept: The degree of an equation/polynomial is the highest exponent of the variable. If that highest exponent is $2$, the equation is called a quadratic equation and is usually written as $ax^2+bx+c=0$. Option A is wrong because linear equations have degree $1$. Option C is wrong because cubic equations have degree $3$. Option D refers to a constant (no variable), which has degree $0$. Exam tip: Always simplify and write the expression in standard polynomial form, then read off the largest exponent to determine the degree.

Question 21

Which of the following expressions clearly contains the $x^2$ term?

Accepted Answer

$4x^2+1=0$ A quadratic term is the term where the variable has exponent 2, i.e. contains $x^2$. Option C explicitly has $4x^2$, so it is correct. The other options contain $10x$, $2x$ and $x$, which are linear (degree 1) terms and not quadratic. The closest possible confusion could be mistaking a large coefficient (like 10 in option A) for a squared term, but the exponent must be 2. Exam tip: always check the exponent on x — only an explicit $x^2$ (or equivalent) makes the term quadratic.

Question 22

What is the linear term in the equation $9x^2+0x+2=0$?

Accepted Answer

0x The linear term is the term containing only $x$ (power 1). In the standard form $ax^2+bx+c$, the linear term is $bx$. Here $b=0$, so the linear term is $0x$. The closest distractor, $9x$, would be correct only if $b=9$, which is not the case; $9x^2$ is the quadratic term and $2$ is the constant term. Exam tip: write the quadratic as $ax^2+bx+c$ and read off $b$; if $b=0$, the linear term is $0x$ or is omitted in the written expression.

Question 23

Which of the following equations is equivalent to $x^2-6x+9=0$?

Accepted Answer

$(x-3)^2=0$ Since $(x-3)^2 = x^2-6x+9$, the equation $(x-3)^2=0$ is equivalent to the given quadratic. It has a repeated root $x=3$. The closest distractor $(x+3)^2=0$ expands to $x^2+6x+9$, so the sign of the linear term is wrong and that option is incorrect. Exam tip: Recognise perfect-square trinomials by checking if the constant equals the square of half the linear coefficient (here $(-6/2)^2=9$).

Question 24

How do we write $x^2-49=0$ in factorised (factor) form?

Accepted Answer

$(x-7)(x+7)=0$ $x^2-49$ is a difference of squares. Using $a^2-b^2=(a-b)(a+b)$ with $a=x, b=7$ gives $(x-7)(x+7)$, so the factorised equation is $(x-7)(x+7)=0$. Option B expands to $x^2-48x-49$, so its middle term is incorrect. Options C and D expand to $x^2+14x+49$ and $x^2-14x+49$ respectively, which do not match the original expression. Exam tip: recognize the difference-of-squares pattern quickly and apply $(a-b)(a+b)$.

Question 25

If $x=2$, what is the left-hand side of the equation $x^2+5x-14=0$?

Accepted Answer

0 Substituting $x=2$ gives $2^2+5\cdot2-14=4+10-14=0$. Hence the left-hand side equals 0, so $x=2$ is a root of the quadratic. Option C (4) is wrong because it corresponds to evaluating only $x^2$; option D (-4) is the value of $5x-14$ alone. Exam tip: substitute values and simplify step by step (evaluate each term first), then combine to avoid sign or arithmetic mistakes.

Introduction to Quadratic Equations

Practice questions

01 In the equation \(7x^2+2x-5=0\), what is the value of \(a\)?

02 What is the value of b in the quadratic equation \(5x^2-9x+4=0\)?

03 In the equation \(8x^2+3x+12=0\), what is the value of \(c\)?

04 In the quadratic equation \(11x^2-6x=0\), what is the value of the constant term \(c\)?

05 In the equation \(13x^2-20=0\), what is the value of \(b\) in the standard quadratic form \(ax^2+bx+c=0\)?

06 Under which condition will the equation \(px^2+qx+r=0\) be called a quadratic equation?

07 What is the standard form of \(4x^2+7x=9\)?

08 What is the standard form of the equation \(2x^2=5x+3\)?

09 If \(x=3\), what is the left-hand side of \(x^2-4x+3=0\)?

10 Which of the following is a solution of the equation \(x^2-25=0\)?

11 What equation do we obtain when we substitute \(x=4\) into \(x^2-8x+16=0\)?

12 For the equation 12x^2 = 0, what are a, b and c respectively?

13 Write \(2x(x+7)=0\) in standard form \(ax^2+bx+c=0\).

14 What is the expanded (standard quadratic) form of \((x+4)(x+5)=0\)?

15 What is the standard form of the equation \((x-6)(x+2)=0\)?

16 What is the coefficient of (x^2) in (x^2+12x+36=0)?

17 In the equation \( -2x^{2}+5x-1=0 \), what is the value of \(a\) according to the standard form \(ax^{2}+bx+c=0\)?

18 Which of the following expresses \(x^2+10x+25=0\) in factorised (product) form?

19 Which of the following is a solution of the quadratic equation \(x^2-3x-10=0\)?

20 If the highest power (degree) of the variable in an equation is \(2\), what is that equation called?

21 Which of the following expressions clearly contains the \(x^2\) term?

22 What is the linear term in the equation \(9x^2+0x+2=0\)?

23 Which of the following equations is equivalent to \(x^2-6x+9=0\)?

24 How do we write \(x^2-49=0\) in factorised (factor) form?

25 If \(x=2\), what is the left-hand side of the equation \(x^2+5x-14=0\)?

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