Q: What is the linear term in the equation $2x^2+0x-7=0$?

$0x$ A linear term is the term where the variable x has exponent 1. The terms in the equation are $2x^2$ (degree 2 — quadratic), $0x$ (degree 1 — linear), and $-7$ (constant). So the linear term is $0x$; its coefficient is zero so it contributes nothing numerically, which is why no visible linear part appears. Why the closest distractor is wrong: $2x$ would be linear, but it is not present in this equation. Exam tip: identify the power of x to find linear terms — look for exponent 1, even if the coefficient is 0.

Q: Which of the following equations is equivalent to $x^2+2x+1=0$?

$(x+1)^2=0$ Expanding $(x+1)^2$ gives $(x+1)^2 = x^2+2x+1$, so $(x+1)^2=0$ is equivalent to the given equation. The closest distractor $(x-1)^2=0$ expands to $x^2-2x+1$, which differs in the sign of the linear term. Exam tip: check if the quadratic is a perfect square by seeing if the middle coefficient equals twice some number (here 2 = 2·1), then write it as $(x±1)^2$ to solve quickly.

Q: Write $x^2-16=0$ in factorised (factor) form.

$(x-4)(x+4)=0$ $x^2-16$ is a difference of squares since $16=4^2$. Use the identity $a^2-b^2=(a-b)(a+b)$ with $a=x$ and $b=4$ to get $(x-4)(x+4)=0$. The other choices give different middle terms when expanded; for example $(x-4)^2$ yields a $-8x$ term, so it does not match the original expression. Exam tip: spot the difference-of-squares pattern or expand candidate factors to check coefficients quickly.

Q: If $x=1$, what is the value of the left-hand side of $x^2+2x-3=0$?

$0$ Substitute $x=1$: left-hand side is $1^2+2\cdot1-3=1+2-3=0$. Hence the left-hand side equals $0$, so option A is correct. The closest distractor (option B = $1$) is wrong because the arithmetic gives $0$, not $1$. Exam tip: when substituting, evaluate powers first, then multiplications, then additions/subtractions step by step to avoid simple mistakes.

Q: In the equation $7x^2-14=0$, what are the coefficients (a), (b) and (c)?

(7, 0, -14) Put the equation in standard form $ax^2+bx+c=0$: $7x^2+0x-14=0$. Thus $a=7,\ b=0,\ c=-14$. Options B and D are wrong because they swap the positions of b and c (or a and c), which changes the coefficients. Exam tip: explicitly write any missing term as $0x$ to identify a, b and c correctly.

Q: What is the standard form of $3x(x-2)=0$?

$3x^2-6x=0$ Distribute: $3x(x-2)=3x\cdot x-3x\cdot 2=3x^2-6x$. Thus the standard form is $3x^2-6x=0$. The closest distractor $3x^2+6x=0$ simply has the sign wrong. Exam tip: expand using the distributive property and collect terms into $ax^2+bx+c=0$; include $+0$ if the constant term is zero to make the form explicit.

Q: If $p(x)=x^2-1$, what is $p(1)$?

0 Compute: $p(1)=1^2-1=0$. So the value is 0. When $p(a)=0$, the number $a$ is called a zero (root) of the polynomial. Choice C (−1) is a common slip — students often mishandle the subtraction; choices B (1) and D (2) result from simple arithmetic errors. Exam tip: substitute values carefully, evaluate powers before subtraction, and recheck your arithmetic.

Q: In the equation $2x^2+3x+1=0$, which term is the constant term?

1 The constant term is the term that does not contain $x$ (equivalently the coefficient of $x^0$). In $2x^2+3x+1=0$, only 1 has no $x$, so 1 is the constant term. $2x^2$ is the quadratic term and $3x$ is the linear term — both contain $x$. Option 0 is incorrect because 0 is not a term present in the polynomial. Exam tip: look for the term with $x$ raised to the power 0 to find the constant term.

Q: Which term is absent in the equation $x^2+10x=0$?

Constant term Write a quadratic in standard form $ax^2+bx+c=0$. The given equation is $x^2+10x+0=0$, so $a=1, b=10, c=0$. The constant term $c$ is zero and therefore absent. The quadratic term $x^2$ and linear term $10x$ are present, so options A and B are incorrect. Exam tip: always rewrite the polynomial as $ax^2+bx+c=0$ and read off $a,b,c$ to spot any missing term quickly.

Question 1

What is the linear term in the equation $2x^2+0x-7=0$?

Accepted Answer

$0x$ A linear term is the term where the variable x has exponent 1. The terms in the equation are $2x^2$ (degree 2 — quadratic), $0x$ (degree 1 — linear), and $-7$ (constant). So the linear term is $0x$; its coefficient is zero so it contributes nothing numerically, which is why no visible linear part appears. Why the closest distractor is wrong: $2x$ would be linear, but it is not present in this equation. Exam tip: identify the power of x to find linear terms — look for exponent 1, even if the coefficient is 0.

Question 2

Which of the following equations is equivalent to $x^2+2x+1=0$?

Accepted Answer

$(x+1)^2=0$ Expanding $(x+1)^2$ gives $(x+1)^2 = x^2+2x+1$, so $(x+1)^2=0$ is equivalent to the given equation. The closest distractor $(x-1)^2=0$ expands to $x^2-2x+1$, which differs in the sign of the linear term. Exam tip: check if the quadratic is a perfect square by seeing if the middle coefficient equals twice some number (here 2 = 2·1), then write it as $(x±1)^2$ to solve quickly.

Question 3

Write $x^2-16=0$ in factorised (factor) form.

Accepted Answer

$(x-4)(x+4)=0$ $x^2-16$ is a difference of squares since $16=4^2$. Use the identity $a^2-b^2=(a-b)(a+b)$ with $a=x$ and $b=4$ to get $(x-4)(x+4)=0$. The other choices give different middle terms when expanded; for example $(x-4)^2$ yields a $-8x$ term, so it does not match the original expression. Exam tip: spot the difference-of-squares pattern or expand candidate factors to check coefficients quickly.

Question 4

If $x=1$, what is the value of the left-hand side of $x^2+2x-3=0$?

Accepted Answer

$0$ Substitute $x=1$: left-hand side is $1^2+2\cdot1-3=1+2-3=0$. Hence the left-hand side equals $0$, so option A is correct. The closest distractor (option B = $1$) is wrong because the arithmetic gives $0$, not $1$. Exam tip: when substituting, evaluate powers first, then multiplications, then additions/subtractions step by step to avoid simple mistakes.

Question 5

In the equation $7x^2-14=0$, what are the coefficients (a), (b) and (c)?

Accepted Answer

(7, 0, -14) Put the equation in standard form $ax^2+bx+c=0$: $7x^2+0x-14=0$. Thus $a=7,\ b=0,\ c=-14$. Options B and D are wrong because they swap the positions of b and c (or a and c), which changes the coefficients. Exam tip: explicitly write any missing term as $0x$ to identify a, b and c correctly.

Question 6

Which of the following equations is quadratic in x but not in y?

Accepted Answer

x^2 + 5x + 6 = 0 In option (A) the highest power of x is 2 (the term x^2), so the equation is quadratic in x. y does not appear, so it is not quadratic in y (degree in y = 0). Option (C) might misleadingly look correct because it contains x^2, but it also contains y^2, so it is quadratic in both x and y and hence not the required answer. Option (D) is linear in x. Exam tip: Determine the degree with respect to a specific variable by checking the highest exponent of that variable; an absent variable has degree 0 and is not quadratic.

Question 7

What is the standard form of $3x(x-2)=0$?

Accepted Answer

$3x^2-6x=0$ Distribute: $3x(x-2)=3x\cdot x-3x\cdot 2=3x^2-6x$. Thus the standard form is $3x^2-6x=0$. The closest distractor $3x^2+6x=0$ simply has the sign wrong. Exam tip: expand using the distributive property and collect terms into $ax^2+bx+c=0$; include $+0$ if the constant term is zero to make the form explicit.

Question 8

If $p(x)=x^2-1$, what is $p(1)$?

Accepted Answer

0 Compute: $p(1)=1^2-1=0$. So the value is 0. When $p(a)=0$, the number $a$ is called a zero (root) of the polynomial. Choice C (−1) is a common slip — students often mishandle the subtraction; choices B (1) and D (2) result from simple arithmetic errors. Exam tip: substitute values carefully, evaluate powers before subtraction, and recheck your arithmetic.

Question 9

In the equation $2x^2+3x+1=0$, which term is the constant term?

Accepted Answer

1 The constant term is the term that does not contain $x$ (equivalently the coefficient of $x^0$). In $2x^2+3x+1=0$, only 1 has no $x$, so 1 is the constant term. $2x^2$ is the quadratic term and $3x$ is the linear term — both contain $x$. Option 0 is incorrect because 0 is not a term present in the polynomial. Exam tip: look for the term with $x$ raised to the power 0 to find the constant term.

Question 10

Which term is absent in the equation $x^2+10x=0$?

Accepted Answer

Constant term Write a quadratic in standard form $ax^2+bx+c=0$. The given equation is $x^2+10x+0=0$, so $a=1, b=10, c=0$. The constant term $c$ is zero and therefore absent. The quadratic term $x^2$ and linear term $10x$ are present, so options A and B are incorrect. Exam tip: always rewrite the polynomial as $ax^2+bx+c=0$ and read off $a,b,c$ to spot any missing term quickly.

Question 11

Which equation is a pure quadratic like $x^2+5=0$?

Accepted Answer

$x^2-9=0$ A pure quadratic has only the $x^2$ term and a constant term, with the coefficient of the linear $x$ term equal to zero. Option A, $x^2-9=0$, contains $x^2$ and a constant and has no $x$ term, so it is a pure quadratic. Option B contains a $3x$ term (linear), so it is not pure; option C is linear and option D is cubic. Exam tip: check the coefficient of $x$; for a pure quadratic it must be zero.

Question 12

What happens when $x=-2$ is substituted into the equation $x^{2}+4x+4=0$?

Accepted Answer

The equation is satisfied Substituting gives: $(-2)^2+4(-2)+4 = 4-8+4 = 0$. The left-hand side equals 0, so the equation is satisfied and $x=-2$ is a solution. Option B is incorrect because the equation is not false; options C and D are wrong since the left side is not 4 or -4 but 0. Exam tip: When squaring negatives, remember $(-a)^2= a^2$; evaluate each term carefully when substituting.

Question 13

Given the standard quadratic form $ax^2+bx+c=0$, which equation corresponds to $a=2,\ b=-5,\ c=3$?

Accepted Answer

$2x^2-5x+3=0$ Substitute the given values into the standard form: with a=2, b=-5, c=3 we get $2x^2-5x+3=0$, so option A is correct. Option B is incorrect because it has b = +5 (wrong sign). Option C swaps the values of b and c (b would be -3 there), and option D changes the positions of the coefficients (a would be 5, not 2). Exam tip: always match each coefficient with its position (a with $x^2$, b with $x$, c constant) and check the signs carefully.

Question 14

If $a=1$, $b=0$, $c=-25$, what is the quadratic equation in standard form?

Accepted Answer

$x^2-25=0$ The standard form is $ax^2+bx+c=0$. Substituting gives $1\cdot x^2+0\cdot x+(-25)=0$, which simplifies to $x^2-25=0$. Option B has the wrong sign for the constant term ($+25$ instead of $-25$), option C wrongly assigns $b=25$ and changes $c$, and option D changes/scales the coefficients. Exam tip: Plug values into $ax^2+bx+c=0$ and simplify term-by-term; always check signs.

Question 15

Which equation becomes x² − 7x + 10 = 0 after writing it in standard form?

Accepted Answer

x² − 7x = −10 The governing algebraic rule is that a term moved from one side of an equation to the other changes its sign. Start with the target standard form x² − 7x + 10 = 0. Move +10 from the left side to the right side; it becomes −10, giving x² − 7x = −10. Reversing this operation confirms the result: moving −10 back to the left changes it to +10 and produces x² − 7x + 10 = 0. Option B would give x² − 7x − 10 = 0 after rearrangement, so its constant sign is wrong. Options C and D have +7x rather than −7x, so their linear terms are also incorrect. Hence option A is correct.

Question 16

What is the maximum number of real solutions possible for (x^2+3x+2=0)?

Accepted Answer

(2) A quadratic equation can have at most (2) real solutions. In easy questions, degree indicates the maximum possible solutions.

Question 17

For the equation $4x^2+4x+1=0$, if the quadratic is written as $ax^2+bx+c=0$, what is the value of $a+b+c$?

Accepted Answer

9 Match the equation to the standard form $ax^2+bx+c=0$: here $a=4$, $b=4$, $c=1$. So $a+b+c=4+4+1=9$. Option B (8) arises from the common mistake of omitting the constant term $c$ and adding only $4+4=8$; therefore 8 is incorrect. Exam tip: always write the quadratic in the form $ax^2+bx+c=0$ first, then read off and add the coefficients carefully.

Question 18

For the quadratic equation $2x^2-3x-5=0$, what is the value of $a-b+c$? (Here $a,b,c$ are the coefficients in $ax^2+bx+c=0$.)

Accepted Answer

0 From the standard form $ax^2+bx+c=0$ we have $a=2$, $b=-3$, $c=-5$. Substituting gives $a-b+c=2-(-3)+(-5)=2+3-5=0$. Common mistake: treating $b$ as +3 yields $2-3-5=-6$, which is incorrect. Exam tip: always write the signs of coefficients explicitly before substituting to avoid sign errors.

Question 19

What is the degree of the equation $5x^2+2x-1=0$?

Accepted Answer

2 The left-hand side is the polynomial $5x^2+2x-1$. The highest power of x present is 2 (from the term $x^2$), so the degree of the equation is 2 — it is a quadratic equation. Option B (degree 1) would indicate a linear equation and is therefore incorrect; option A (0) refers to a constant polynomial. Exam tip: simplify the expression first if needed and then take the largest exponent of the variable (ignore terms with zero coefficients).

Question 20

Which of the following is the factorised form of $x^2-2x-15=0$?

Accepted Answer

$(x-5)(x+3)=0$ Expand the factors in A: $(x-5)(x+3)=x^2+3x-5x-15=x^2-2x-15$, so A matches the given quadratic. A common distractor B expands to $(x+5)(x-3)=x^2+2x-15$ which has +2x (not -2x), so it is incorrect. Exam tip: find two numbers whose product is -15 and sum is -2 (here -5 and 3) to factor quickly.

Question 21

A rectangle has length $x+3$ and width $x$. If its area is $10$, which equation represents this situation?

Accepted Answer

$x(x+3)=10$ Area = length × width. With length $x+3$ and width $x$, the equation is $x(x+3)=10$, which rearranges to the quadratic $x^2+3x-10=0$ for solving. Option C ($2x+3=10$) might come from a mistaken linear relation (not the area formula); options B and D do not represent the product of length and width. Exam tip: Always form the product first, then bring all terms to one side to convert to standard quadratic form before solving.

Question 22

The product of two consecutive positive integers is 30. If the smaller integer is $x$, which of the following equations represents this situation?

Accepted Answer

$x(x+1)=30$ If the smaller integer is $x$, the next consecutive integer is $x+1$. Since the relationship given is product, the correct equation is $x(x+1)=30$. Option B uses a sum $x+x+1=30$ instead of a product, so it is incorrect. Option C represents $x$ being the larger integer (product $x(x-1)$), which contradicts the statement that $x$ is the smaller one. Option D, $2x=30$, models two equal integers ($x+x$), not consecutive integers. Exam tip: carefully note whether the problem states "product" or "sum" and which integer (smaller/larger) is denoted by the variable.

Question 23

Which option shows the common mistake when converting the equation $x^2 = 5x - 6$ to standard form?

Accepted Answer

$x^2 - 5x - 6 = 0$ The correct standard form is $x^2 - 5x + 6 = 0$. To convert, bring all terms to one side and set the equation equal to zero; moving $-6$ from the right to the left changes its sign to $+6$. Option (B) shows the common mistake where the sign of $-6$ was not changed, so it is the wrong conversion and therefore the required 'common mistake'. The closest distractor (D) simply leaves the equation unchanged and is not in standard form; (C) unnecessarily writes $0x$, which is not the standard conversion. Exam tip: always move all terms to one side, flip signs when moving terms, combine like terms and set equal to zero.

Question 24

Which equation has (x=0) as a solution?

Accepted Answer

(x^2+5x=0) Putting (x=0) in option (A) gives (0=0). Direct substitution is the simplest way to check a solution.

Question 25

If x² + kx + 12 = 0 is a quadratic equation, then k is the coefficient of which term?

Accepted Answer

The x term The governing concept is comparison with the standard quadratic form ax² + bx + c = 0. In this form, a is the coefficient of x², b is the coefficient of x, and c is the constant term. In the given equation, the middle term is kx. Since the coefficient is the factor multiplying the variable, k is the coefficient of x in kx. The coefficient of x² is 1, while 12 is the constant term because it contains no variable. The zero on the right is the value to which the expression is equated, not the term represented by k. Therefore, k is the coefficient of the x term, so option B is correct. The other choices confuse the roles of a, b, and c.

Introduction to Quadratic Equations

Practice questions

01 What is the linear term in the equation \(2x^2+0x-7=0\)?

02 Which of the following equations is equivalent to \(x^2+2x+1=0\)?

03 Write \(x^2-16=0\) in factorised (factor) form.

04 If \(x=1\), what is the value of the left-hand side of \(x^2+2x-3=0\)?

05 In the equation \(7x^2-14=0\), what are the coefficients (a), (b) and (c)?

06 Which of the following equations is quadratic in x but not in y?

07 What is the standard form of \(3x(x-2)=0\)?

08 If \(p(x)=x^2-1\), what is \(p(1)\)?

09 In the equation \(2x^2+3x+1=0\), which term is the constant term?

10 Which term is absent in the equation \(x^2+10x=0\)?

11 Which equation is a pure quadratic like \(x^2+5=0\)?

12 What happens when \(x=-2\) is substituted into the equation \(x^{2}+4x+4=0\)?

13 Given the standard quadratic form \(ax^2+bx+c=0\), which equation corresponds to \(a=2,\ b=-5,\ c=3\)?

14 If \(a=1\), \(b=0\), \(c=-25\), what is the quadratic equation in standard form?

15 Which equation becomes x² − 7x + 10 = 0 after writing it in standard form?

16 What is the maximum number of real solutions possible for (x^2+3x+2=0)?

17 For the equation \(4x^2+4x+1=0\), if the quadratic is written as \(ax^2+bx+c=0\), what is the value of \(a+b+c\)?

18 For the quadratic equation \(2x^2-3x-5=0\), what is the value of \(a-b+c\)? (Here \(a,b,c\) are the coefficients in \(ax^2+bx+c=0\).)

19 What is the degree of the equation \(5x^2+2x-1=0\)?

20 Which of the following is the factorised form of \(x^2-2x-15=0\)?

21 A rectangle has length \(x+3\) and width \(x\). If its area is \(10\), which equation represents this situation?

22 The product of two consecutive positive integers is 30. If the smaller integer is \(x\), which of the following equations represents this situation?

23 Which option shows the common mistake when converting the equation \(x^2 = 5x - 6\) to standard form?

24 Which equation has (x=0) as a solution?

25 If x² + kx + 12 = 0 is a quadratic equation, then k is the coefficient of which term?

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