Q: In the equation $x^2-5x+6=0$, matching it with the standard form $ax^2+bx+c=0$, what are the values of $a, b, c$?

(a=1, b=-5, c=6) Compare with the standard form $ax^2+bx+c=0$. Here the coefficient of $x^2$ is 1 so $a=1$; the coefficient of $x$ is -5 so $b=-5$; the constant term is 6 so $c=6$. Option B is a common sign-mistake (it shows $b=5$ instead of $-5$). Option C incorrectly negates $a$, and D has the terms permuted. Exam tip: always identify the coefficient of $x^2$ first and copy signs exactly from the equation.

Q: Is $x=2$ a root of the equation $x^2-4=0$?

Yes Substituting $x=2$ gives $2^2-4=4-4=0$, so $x=2$ is indeed a root. Option C is incorrect because $x=-2$ also satisfies the equation ($(-2)^2-4=0$), so it is false to claim 'only $x=-2$' is a root — both $2$ and $-2$ are roots. Exam tip: To check a candidate root, substitute it into the polynomial and verify the result equals $0$.

Q: The square of a number (x) plus (3) times the number is (10). Which equation is correct?

(x^2+3x=10) Square means (x^2) and (3) times means (3x). In word problems, convert the sentence slowly into an equation.

Q: Why is 3x + 2 = 0 not a quadratic equation?

Because the highest power is 1 The governing concept is the degree of an equation. A quadratic equation must contain a variable whose highest power is exactly 2, and the coefficient of that squared term must be non-zero. In 3x + 2 = 0, the variable x occurs only as x¹, so the highest power is 1 and the equation has degree 1. It is therefore a linear equation. The constant term 2 does not make an equation quadratic; linear equations can also contain constant terms. Option A is incorrect because there is no x² term at all. Option C is incorrect because x is clearly present. Hence option D correctly explains why the equation is not quadratic.

Q: What is the maximum number of roots of a quadratic equation?

(2) A quadratic equation has degree (2), so it can have at most (2) roots. Remember degree and maximum number of roots together.

Q: What is the coefficient of x² in 5x² − 2x + 1 = 0?

5 The governing concept is identification of coefficients in a polynomial or quadratic expression. A coefficient is the numerical factor that multiplies a variable term. In 5x² − 2x + 1 = 0, the term containing x² is 5x², so the coefficient of x² is 5. The negative sign before 2 belongs to the term −2x and therefore describes the coefficient of x, not the coefficient of x². The number 1 is the constant term, so it cannot be the requested coefficient. Option B ignores the sign and refers to the x-term, option C is the constant, and option D is the coefficient of x. Therefore option A is the only correct answer.

Q: What is the standard form of the equation $4 = 3x - x^2$?

x^2-3x+4=0 To write in standard form bring all terms to one side as $ax^2+bx+c=0$. From $4 = 3x - x^2$, moving the right-hand terms to the left changes signs: $-x^2$ becomes $+x^2$, $+3x$ becomes $-3x$, and $+4$ stays. So the standard form is $x^2-3x+4=0$. The closest distractor is option C ($x^2-3x-4=0$) which has the constant term sign wrong; option B has the sign of the linear term wrong. Exam tip: always rearrange to $ax^2+bx+c=0$ and check sign changes when moving terms across ‘=’.

Q: What is the linear term in (x^2-7=0)?

(0x) There is no (x) term, so the linear term is considered (0x). A missing term has coefficient (0).

Question 1

What is the degree of the equation 7x² − 4x + 9 = 0?

Accepted Answer

2 Answer: B, 2. The degree of a polynomial equation is the greatest exponent of the variable whose coefficient is not zero. In 7x² − 4x + 9 = 0, the terms are 7x², −4x, and 9. Their powers of x are 2, 1, and 0 respectively. The coefficient of x² is 7, which is non-zero, so the highest actual power is 2. Therefore the equation has degree 2 and is called a quadratic equation. A, 1, describes only the linear term −4x and ignores the higher-power term. C, 3, is not present anywhere in the equation, so it cannot be the degree. D, 0, is the degree of a non-zero constant polynomial, not of this expression. The equals-zero sign does not change the degree; we inspect the polynomial on the left. Memory cue: degree means the largest exponent that remains after ignoring zero-coefficient terms.

Question 2

In the equation $x^2-5x+6=0$, matching it with the standard form $ax^2+bx+c=0$, what are the values of $a, b, c$?

Accepted Answer

(a=1, b=-5, c=6) Compare with the standard form $ax^2+bx+c=0$. Here the coefficient of $x^2$ is 1 so $a=1$; the coefficient of $x$ is -5 so $b=-5$; the constant term is 6 so $c=6$. Option B is a common sign-mistake (it shows $b=5$ instead of $-5$). Option C incorrectly negates $a$, and D has the terms permuted. Exam tip: always identify the coefficient of $x^2$ first and copy signs exactly from the equation.

Question 3

For the quadratic equation $ax^2+bx+c=0$, what condition must the coefficient $a$ satisfy?

Accepted Answer

a \neq 0 A quadratic must have a nonzero coefficient for the highest power $x^2$. If $a=0$, the $x^2$ term disappears and the equation reduces to the linear form $bx+c=0$. Thus the required condition is $a\neq 0$. Option B ($a<0$) is incorrect because $a$ need not be negative; it can be positive. Option D ($a=1$) is only a special case, not a necessity. Exam tip: to confirm the degree of a polynomial, always check that the coefficient of the highest-power term is nonzero.

Question 4

Is $x=2$ a root of the equation $x^2-4=0$?

Accepted Answer

Yes Substituting $x=2$ gives $2^2-4=4-4=0$, so $x=2$ is indeed a root. Option C is incorrect because $x=-2$ also satisfies the equation ($(-2)^2-4=0$), so it is false to claim 'only $x=-2$' is a root — both $2$ and $-2$ are roots. Exam tip: To check a candidate root, substitute it into the polynomial and verify the result equals $0$.

Question 5

The square of a number (x) plus (3) times the number is (10). Which equation is correct?

Accepted Answer

(x^2+3x=10) Square means (x^2) and (3) times means (3x). In word problems, convert the sentence slowly into an equation.

Question 6

Why is 3x + 2 = 0 not a quadratic equation?

Accepted Answer

Because the highest power is 1 The governing concept is the degree of an equation. A quadratic equation must contain a variable whose highest power is exactly 2, and the coefficient of that squared term must be non-zero. In 3x + 2 = 0, the variable x occurs only as x¹, so the highest power is 1 and the equation has degree 1. It is therefore a linear equation. The constant term 2 does not make an equation quadratic; linear equations can also contain constant terms. Option A is incorrect because there is no x² term at all. Option C is incorrect because x is clearly present. Hence option D correctly explains why the equation is not quadratic.

Question 7

What is the maximum number of roots of a quadratic equation?

Accepted Answer

(2) A quadratic equation has degree (2), so it can have at most (2) roots. Remember degree and maximum number of roots together.

Question 8

What is the coefficient of x² in 5x² − 2x + 1 = 0?

Accepted Answer

5 The governing concept is identification of coefficients in a polynomial or quadratic expression. A coefficient is the numerical factor that multiplies a variable term. In 5x² − 2x + 1 = 0, the term containing x² is 5x², so the coefficient of x² is 5. The negative sign before 2 belongs to the term −2x and therefore describes the coefficient of x, not the coefficient of x². The number 1 is the constant term, so it cannot be the requested coefficient. Option B ignores the sign and refers to the x-term, option C is the constant, and option D is the coefficient of x. Therefore option A is the only correct answer.

Question 9

What is the standard form of the equation $4 = 3x - x^2$?

Accepted Answer

x^2-3x+4=0 To write in standard form bring all terms to one side as $ax^2+bx+c=0$. From $4 = 3x - x^2$, moving the right-hand terms to the left changes signs: $-x^2$ becomes $+x^2$, $+3x$ becomes $-3x$, and $+4$ stays. So the standard form is $x^2-3x+4=0$. The closest distractor is option C ($x^2-3x-4=0$) which has the constant term sign wrong; option B has the sign of the linear term wrong. Exam tip: always rearrange to $ax^2+bx+c=0$ and check sign changes when moving terms across ‘=’.

Question 10

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

Accepted Answer

(0x) There is no (x) term, so the linear term is considered (0x). A missing term has coefficient (0).

Question 11

What is the constant term in $2x^2+3x-8=0$?

Accepted Answer

-8 The constant term is the term that does not contain the variable x. In $2x^2+3x-8$ the only term without x is $-8$, so the constant term is $-8$. Option C (8) is a common distractor but incorrect because the sign is negative. Exam tip: to find the constant quickly, scan for the term with no x and pay attention to its sign.

Question 12

Which is the quadratic term in the equation $6x^2 - x + 5 = 0$?

Accepted Answer

6x^2 A quadratic term is the term in which the variable has exponent 2 (i.e. contains $x^2$). In the equation $6x^2 - x + 5 = 0$, only $6x^2$ contains $x^2$, so it is the quadratic term. The distractors $-x$ and $x$ are linear terms (exponent 1) and $5$ is a constant (exponent 0). Exam tip: check the exponent of the variable — the term with $x^2$ is the quadratic term.

Question 13

Which of the following is a pure quadratic equation?

Accepted Answer

$4x^2-9=0$ A pure quadratic equation has the form $ax^2+bx+c=0$ with the linear coefficient $b=0$, so it reduces to $ax^2+c=0$. In $4x^2-9=0$ there is no $x$ term, so it is a pure quadratic. The closest distractor, $x^2+2x+1=0$, is a quadratic but not pure because of the $2x$ term. Exam tip: check the coefficient of $x$; if it is zero (or the $x$ term is absent) the equation is a pure quadratic.

Question 14

Which of the following is a monic quadratic equation?

Accepted Answer

$x^2+5x+6=0$ A monic quadratic has the coefficient of $x^2$ equal to 1. In option A, $x^2+5x+6=0$ has coefficient 1 for $x^2$, so it is monic. The closest distractor is option B because it is also a quadratic but has coefficient 2 for $x^2$, so it is not monic. Exam tip: to identify a monic quadratic quickly, check only the coefficient of $x^2$; if it is 1, the quadratic is monic.

Question 15

Is $x=0$ a root of the equation $x^2+5x=0$?

Accepted Answer

Yes Check by substitution: $0^2+5\cdot0=0$, so $x=0$ makes the equation zero and is therefore a root. Factoring gives $x^2+5x=x(x+5)=0$, so the roots are $x=0$ and $x=-5$. Option C is wrong because $x=5$ yields $25+25\neq0$. Option D is misleading: $x=-5$ is a root but not the only one. Exam tip: verify roots by direct substitution or factor the quadratic (look for a common factor first).

Question 16

What is the discriminant of the quadratic equation $ax^2+bx+c=0$?

Accepted Answer

$b^2-4ac$ The discriminant is denoted by $D$ and is given by $D=b^2-4ac$. It determines the nature of the roots: two distinct real roots if $D>0$, one repeated real root if $D=0$, and two complex conjugate roots if $D<0$. Option B has the wrong sign ($+4ac$ instead of $-4ac$), while options A and D mix up the coefficients and are not the standard discriminant. Exam tip: correctly identify $a,b,c$ from the equation and substitute into $b^2-4ac$ to avoid sign errors.

Question 17

If the discriminant $D=b^2-4ac$ of a quadratic equation $ax^2+bx+c=0$ equals 0, what will be the nature of its roots?

Accepted Answer

Two equal real roots When $D=b^2-4ac=0$, the quadratic has two equal (repeated) real roots. The common root equals $x=-\dfrac{b}{2a}$. Option A is a close distractor — two distinct real roots occur only when $D>0$. Option C is wrong because no real roots occur only when $D<0$. Option D is impossible since a quadratic cannot have three roots. Exam tip: compute $D$ first and compare it with 0 to decide the nature of roots.

Question 18

If the discriminant (D<0), what is true about real roots?

Accepted Answer

It has no real roots When (D<0), there are no real roots. A negative discriminant gives no roots on the real number line.

Question 19

Which are the correct values of (a, b, c) in −x² + 4x − 1 = 0?

Accepted Answer

a = −1, b = 4, c = −1 Answer: A, a = −1, b = 4, c = −1. Compare the equation with the standard quadratic form ax² + bx + c = 0. The coefficient attached to x² is a, the coefficient attached to x is b, and the term without x is c. In −x² + 4x − 1 = 0, the coefficient of x² is −1, so a = −1. The coefficient of x is +4, so b = 4. The constant term is −1, so c = −1. The sign before each term must be carried carefully. B changes the sign of a, even though the first term is negative. C changes the sign of b, although the x-term is positive. D rearranges the roles of the coefficients and constant instead of matching powers of x. Therefore A is the only complete and correct matching. Memory cue: read coefficients by descending powers: x² gives a, x gives b, and the number alone gives c.

Question 20

Why is (0x^2+2x+3=0) not quadratic?

Accepted Answer

Because the coefficient of (x^2) is (0) Here the coefficient of (x^2) is (0), so the (x^2) term disappears. For a quadratic equation, (a
eq 0) is necessary.

Question 21

Which equation is obtained by expanding $(x+1)(x+2)=0$?

Accepted Answer

x^2+3x+2=0 Multiply each term of the first binomial by each term of the second: $(x+1)(x+2)=x\cdot x + x\cdot 2 + 1\cdot x + 1\cdot 2 = x^2+2x+x+2 = x^2+3x+2$. Thus the expanded equation is $x^2+3x+2=0$. The closest distractor $x^2+3x-2=0$ has the constant term sign wrong; the other options have incorrect middle or constant terms. Exam tip: use FOIL and combine like terms carefully; check signs for the constant term.

Question 22

What is the standard form of the equation $x^2 = 7$?

Accepted Answer

$x^2 - 7 = 0$ Standard form means writing all terms on one side as $ax^2+bx+c=0$. For $x^2=7$, move 7 to the left to get $x^2-7=0$, which is the standard form. Option A is incorrect because $x^2+7=0$ corresponds to $x^2=-7$. Option C is wrong since it introduces a $ -7x$ term not present in the original equation. Exam tip: To convert to standard form, bring every term to the left and arrange as $ax^2+bx+c=0$.

Question 23

What are the roots of $x^2-1=0$?

Accepted Answer

(-1, 1) This is a difference of squares: $x^2-1=(x-1)(x+1)$. The equation equals zero when $x-1=0$ or $x+1=0$, so $x=1$ and $x=-1$. Option A (0,1) is incorrect because at $x=0$ we get $0^2-1=-1\neq0$. Options C and D also do not satisfy the equation. Exam tip: recognize $x^2-a^2$ and factor, or take square roots to write $x=\pm a$.

Question 24

What are the roots of $x^2=9$?

Accepted Answer

(-3, 3) From $x^2=9$ we get $x=\pm\sqrt{9}=\pm3$, so the roots are $-3$ and $3$. Option B wrongly lists 9 (although 3 is correct) — but $9^2=81$, not 9. Option D has ±9 whose squares are 81, and option A contains 0 whose square is 0, so none of those satisfy the equation. Exam tip: for equations of the form $x^2=a$ always take both ±√a and verify by substituting back.

Question 25

Which quadratic equation has roots (0) and (-3)?

Accepted Answer

(x^2+3x=0) For roots (0) and (-3), the factors are (x) and (x+3). So the equation is (x(x+3)=0), that is (x^2+3x=0).

Introduction to Quadratic Equations

Practice questions

01 What is the degree of the equation 7x² − 4x + 9 = 0?

02 In the equation \(x^2-5x+6=0\), matching it with the standard form \(ax^2+bx+c=0\), what are the values of \(a, b, c\)?

03 For the quadratic equation \(ax^2+bx+c=0\), what condition must the coefficient \(a\) satisfy?

04 Is \(x=2\) a root of the equation \(x^2-4=0\)?

05 The square of a number (x) plus (3) times the number is (10). Which equation is correct?

06 Why is 3x + 2 = 0 not a quadratic equation?

07 What is the maximum number of roots of a quadratic equation?

08 What is the coefficient of x² in 5x² − 2x + 1 = 0?

09 What is the standard form of the equation \(4 = 3x - x^2\)?

10 What is the linear term in (x^2-7=0)?

11 What is the constant term in \(2x^2+3x-8=0\)?

12 Which is the quadratic term in the equation \(6x^2 - x + 5 = 0\)?

13 Which of the following is a pure quadratic equation?

14 Which of the following is a monic quadratic equation?

15 Is \(x=0\) a root of the equation \(x^2+5x=0\)?

16 What is the discriminant of the quadratic equation \(ax^2+bx+c=0\)?

17 If the discriminant \(D=b^2-4ac\) of a quadratic equation \(ax^2+bx+c=0\) equals 0, what will be the nature of its roots?

18 If the discriminant (D<0), what is true about real roots?

19 Which are the correct values of (a, b, c) in −x² + 4x − 1 = 0?

20 Why is (0x^2+2x+3=0) not quadratic?

21 Which equation is obtained by expanding \((x+1)(x+2)=0\)?

22 What is the standard form of the equation \(x^2 = 7\)?

23 What are the roots of \(x^2-1=0\)?

24 What are the roots of \(x^2=9\)?

25 Which quadratic equation has roots (0) and (-3)?

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