Q: If $x=-5$ is a root of the equation $3x^2+px-20=0$, what is the value of $p$?

11 Since $x=-5$ is a root, substitute it directly into the equation: $3(-5)^2+p(-5)-20=0$. Thus, $75-5p-20=0$, so $55-5p=0$ and $p=11$. The distractor $-11$ results from mishandling the negative sign in $p(-5)$. Exam tip: substitute the given root carefully and simplify before solving for the parameter.

Q: Which monic quadratic equation has sum of roots (-13) and product (42)?

(x^2+13x+42=0) A monic equation is (x^2-(\text{sum})x+\text{product}=0). Substituting sum (-13) gives (x^2+13x+42=0).

Q: If the two roots of the quadratic equation $x^2-2mx+49=0$ are equal, what are the possible values of $m$?

$m=\pm7$ For equal roots, the discriminant $D=b^2-4ac$ must be zero. Here $a=1, b=-2m, c=49$, so $D=(-2m)^2-4(1)(49)=4m^2-196=0$. Thus, $m^2=49$, giving $m=\pm7$. Choosing only $m=7$ or only $m=-7$ omits one valid value. Exam tip: For equal roots of a quadratic equation, set the discriminant directly equal to zero.

Q: What condition on $p$ is necessary and sufficient for the equation $x^2+2px+36=0$ to have real roots?

$p\leq -6$ या $p\geq 6$ A quadratic equation has real roots when its discriminant satisfies $D\geq 0$. Here, $a=1, b=2p, c=36$, so $D=(2p)^2-4(1)(36)=4p^2-144$. Thus, $4p^2-144\geq 0$, which gives $p^2\geq 36$ and hence $p\leq -6$ or $p\geq 6$. In option B, the discriminant is negative, so the roots are not real. Exam tip: include $D=0$, because equal roots are also real roots.

Q: If 7 and 8 are the roots of the equation $x^2-sx+p=0$, what is the value of $s+p$?

71 By Vieta’s formulas, the sum of the roots is $s=7+8=15$, and their product is $p=7\times8=56$. Therefore, $s+p=15+56=71$, so option A is correct. Exam tip: In an equation of the form $x^2-sx+p=0$, the sum of the roots is $s$ and their product is $p$.

Q: What is the sum of squares of the roots of (4x^2+8x+3=0)?

( \frac{5}{2} ) The direct answer is option A, $5/2$. Let the roots of $4x^2+8x+3=0$ be $\alpha$ and $\beta$. Using the root relations, $\alpha+\beta=-8/4=-2$ and $\alpha\beta=3/4$. The identity for the sum of squares is $\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta$. Substitution gives $(-2)^2-2(3/4)=4-3/2=5/2$. Hence option A is correct. Option B, 4, is only the square of the sum and forgets the subtraction of $2\alpha\beta$. Option C, $3/4$, is the product of the roots, not the sum of their squares. Option D, $13/4$, results from an incorrect operation and does not match the identity. The roots themselves need not be solved. Memory cue: square-sum = square of sum − twice product.

Q: If $\alpha$ and $\beta$ are the roots of the equation $x^2-14x+48=0$, what is the value of $(\alpha-6)(\beta-6)$?

0 By Vieta’s formulas, $\alpha+\beta=14$ and $\alpha\beta=48$. Therefore, $(\alpha-6)(\beta-6)=\alpha\beta-6(\alpha+\beta)+36=48-6(14)+36=0$. Hence, option A is correct. Option C is only the value of $\alpha\beta$, not of the complete expression. Exam tip: In such questions, use the sum and product of the roots instead of finding the roots separately.

Question 1

What is the standard form with integer coefficients of (\frac{3x^2+2}{4}-\frac{x-5}{3}=6)?

Accepted Answer

(9x^2-4x-42=0) Multiplying the whole equation by (12) gives (9x^2+6-4x+20=72). Therefore the standard form is (9x^2-4x-46=0).

Question 2

If $x=-5$ is a root of the equation $3x^2+px-20=0$, what is the value of $p$?

Accepted Answer

11 Since $x=-5$ is a root, substitute it directly into the equation: $3(-5)^2+p(-5)-20=0$. Thus, $75-5p-20=0$, so $55-5p=0$ and $p=11$. The distractor $-11$ results from mishandling the negative sign in $p(-5)$. Exam tip: substitute the given root carefully and simplify before solving for the parameter.

Question 3

Which monic quadratic equation has sum of roots (-13) and product (42)?

Accepted Answer

(x^2+13x+42=0) A monic equation is (x^2-(	ext{sum})x+	ext{product}=0). Substituting sum (-13) gives (x^2+13x+42=0).

Question 4

If the two roots of the quadratic equation $x^2-2mx+49=0$ are equal, what are the possible values of $m$?

Accepted Answer

$m=\pm7$ For equal roots, the discriminant $D=b^2-4ac$ must be zero. Here $a=1, b=-2m, c=49$, so $D=(-2m)^2-4(1)(49)=4m^2-196=0$. Thus, $m^2=49$, giving $m=\pm7$. Choosing only $m=7$ or only $m=-7$ omits one valid value. Exam tip: For equal roots of a quadratic equation, set the discriminant directly equal to zero.

Question 5

What condition on $p$ is necessary and sufficient for the equation $x^2+2px+36=0$ to have real roots?

Accepted Answer

$p\leq -6$ या $p\geq 6$ A quadratic equation has real roots when its discriminant satisfies $D\geq 0$. Here, $a=1, b=2p, c=36$, so $D=(2p)^2-4(1)(36)=4p^2-144$. Thus, $4p^2-144\geq 0$, which gives $p^2\geq 36$ and hence $p\leq -6$ or $p\geq 6$. In option B, the discriminant is negative, so the roots are not real. Exam tip: include $D=0$, because equal roots are also real roots.

Question 6

What is the sum of reciprocals of the roots of (5x^2-17x+6=0)?

Accepted Answer

( \frac{17}{6} ) The sum of reciprocals is (\frac{\alpha+\beta}{\alpha\beta}). Here it is (\frac{\frac{17}{5}}{\frac{6}{5}}=\frac{17}{6}).

Question 7

If 7 and 8 are the roots of the equation $x^2-sx+p=0$, what is the value of $s+p$?

Accepted Answer

71 By Vieta’s formulas, the sum of the roots is $s=7+8=15$, and their product is $p=7\times8=56$. Therefore, $s+p=15+56=71$, so option A is correct. Exam tip: In an equation of the form $x^2-sx+p=0$, the sum of the roots is $s$ and their product is $p$.

Question 8

If the roots of the equation $x^2-16x+k=0$ are real and distinct, what is the correct condition on $k$?

Accepted Answer

$k<64$ A quadratic equation has real and distinct roots only when its discriminant is positive. Here, $D=(-16)^2-4(1)(k)=256-4k$. Thus, $256-4k>0$, which gives $k<64$. At $k=64$, the discriminant is zero and the roots are equal, so option B is not correct. Exam tip: For real and distinct roots, always apply the condition $D>0$.

Question 9

Which quadratic equation has a discriminant of −35?

Accepted Answer

x² + x + 9 = 0 For a quadratic equation ax² + bx + c = 0, the discriminant is D = b² − 4ac. In option A, a = 1, b = 1, and c = 9, so D = 1² − 4 × 1 × 9 = 1 − 36 = −35. For example, option C gives 2² − 4 × 3 × 3 = −32, so it is not correct. Exam tip: the sign of b disappears when calculating b², but the complete value of 4ac must be subtracted.

Question 10

What is the sum of squares of the roots of (4x^2+8x+3=0)?

Accepted Answer

( \frac{5}{2} ) The direct answer is option A, $5/2$. Let the roots of $4x^2+8x+3=0$ be $\alpha$ and $\beta$. Using the root relations, $\alpha+\beta=-8/4=-2$ and $\alpha\beta=3/4$. The identity for the sum of squares is $\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta$. Substitution gives $(-2)^2-2(3/4)=4-3/2=5/2$. Hence option A is correct. Option B, 4, is only the square of the sum and forgets the subtraction of $2\alpha\beta$. Option C, $3/4$, is the product of the roots, not the sum of their squares. Option D, $13/4$, results from an incorrect operation and does not match the identity. The roots themselves need not be solved. Memory cue: square-sum = square of sum − twice product.

Question 11

If $\alpha$ and $\beta$ are the roots of the equation $x^2-14x+48=0$, what is the value of $(\alpha-6)(\beta-6)$?

Accepted Answer

0 By Vieta’s formulas, $\alpha+\beta=14$ and $\alpha\beta=48$. Therefore, $(\alpha-6)(\beta-6)=\alpha\beta-6(\alpha+\beta)+36=48-6(14)+36=0$. Hence, option A is correct. Option C is only the value of $\alpha\beta$, not of the complete expression. Exam tip: In such questions, use the sum and product of the roots instead of finding the roots separately.

Question 12

What is the standard form of the equation (x+4)(x+7)+(x−3)(x+6)=88?

Accepted Answer

2x²+14x−78=0 Expand the two products: (x+4)(x+7)=x²+11x+28 and (x−3)(x+6)=x²+3x−18. Their sum is 2x²+14x+10, so the equation becomes 2x²+14x+10=88. Bringing 88 to the left gives 2x²+14x−78=0, so option A is correct. Option B stops before subtracting 88. Exam tip: To write a quadratic equation in standard form, bring all terms to one side so that the other side is 0.

Question 13

If the roots of a monic quadratic equation are (2r) and (5r), and their sum is (21), what will be the constant term?

Accepted Answer

(90) The direct answer is option A: 90. A monic quadratic has leading coefficient 1, so if its roots are r₁ and r₂, its equation is x² - (r₁ + r₂)x + r₁r₂ = 0. Therefore the constant term equals the product of the roots. The roots here are 2r and 5r, and their sum is 21. Thus 2r + 5r = 21, so 7r = 21 and r = 3. The roots are therefore 2(3) = 6 and 5(3) = 15. Their product is 6 × 15 = 90, so the constant term is 90. Option A is correct. Option B, 21, is the sum of the roots, not their product. Option C, 45, does not equal the product 6 × 15 and may result from an incomplete calculation. Option D, 63, is another incorrect product or arithmetic value; it is not the constant term. The word “monic” is important: because the leading coefficient is 1, the constant term directly equals the root product. Memory cue: first find r from the sum, then multiply the two roots.

Question 14

Which statement correctly describes the nature of the roots of $x^2-10x+29=0$?

Accepted Answer

It has no real roots Here, $a=1$, $b=-10$, and $c=29$. The discriminant is $D=b^2-4ac=(-10)^2-4(1)(29)=100-116=-16$. Since $D<0$, the equation has no real roots; its roots are complex. Option B would be correct only if $D=0$, which is not the case here. Exam tip: For a quadratic equation, $D<0$ always indicates that there are no real roots.

Question 15

Which of the following becomes a quadratic equation in the variable $x$ after simplification?

Accepted Answer

$x(x-3)=4$ Expanding $x(x-3)=4$ gives $x^2-3x-4=0$, whose highest power of $x$ is 2, so it is quadratic. Option B is an identity because both sides are identical. Exam tip: bring all terms to one side, then check the highest power.

Question 16

What is the difference between the larger and smaller roots of $5x^2-22x+24=0$?

Accepted Answer

$\frac{2}{5}$ Factoring gives $5x^2-22x+24=(5x-12)(x-2)$. Therefore, the roots are $\frac{12}{5}$ and $2$. The difference between the larger and smaller roots is $\frac{12}{5}-2=\frac{2}{5}$. The values $\frac{12}{5}$ and $2$ are the roots themselves, not their difference. In an exam, subtract the smaller root from the larger root when the difference between roots is asked.

Question 17

If the quadratic equation $x^2-18x+c=0$ can be written as a perfect square, what is the value of $c$?

Accepted Answer

81 For the expression to be a perfect square, $x^2-18x+c$ must equal $(x-9)^2=x^2-18x+81$. Therefore, $c=81$. Choosing 324 incorrectly amounts to squaring 18 directly; the correct method is to square half the coefficient of $x$. Exam tip: $x^2+bx+c$ is a perfect square when $c=\left(\frac{b}{2}\right)^2$.

Question 18

If the two roots of the equation $x^2+bx+100=0$ are equal and each root is $-10$, what is the value of $b$?

Accepted Answer

20 By Vieta’s formula, the sum of the roots of $x^2+bx+100=0$ is $-b$. Since both roots are $-10$, their sum is $-10+(-10)=-20$. Therefore, $-b=-20$, giving $b=20$. Exam tip: for $x^2+bx+c=0$, the sum of the roots is always $-b$; missing this negative sign leads to option B.

Question 19

In which option will the sum of roots be negative and the product of roots be negative?

Accepted Answer

(x^2+5x-24=0) The answer is option A. For a quadratic equation ax^2+bx+c=0, the sum of roots is -b/a and their product is c/a. In option A, a=1, b=5, c=-24, so the sum is -5 and the product is -24; both are negative. Option B has sum 5, although its product is -24, so it fails the first condition. Option C has sum -5, but its product is 24, so it fails the second condition. Option D has sum 5 and product 24, so neither required sign is obtained. Therefore only A works. Remember: the middle coefficient is negated for the sum, but the constant term keeps its sign for the product.

Question 20

A rectangle has length $x+9$ units and breadth $x-6$ units. If its area is 130 square units, which quadratic equation is formed?

Accepted Answer

$x^2+3x-184=0$ Area of a rectangle = length × breadth, so $(x+9)(x-6)=130$. Expanding gives $x^2+3x-54=130$, and rearranging produces $x^2+3x-184=0$. Option B does not correctly account for the $-54$ term, while options C and D contain errors in expansion. Exam tip: multiply the binomials first, then bring all terms to one side and equate the expression to zero.

Question 21

Two numbers have sum (17) and product (72). They can be roots of which quadratic equation?

Accepted Answer

(x^2-17x+72=0) If the sum of roots is (17) and product is (72), the equation is (x^2-17x+72=0). Remember the monic form formula.

Question 22

If (\alpha,\beta) are roots of (x^2-13x+42=0), what is (\frac{1}{\alpha}+\frac{1}{\beta})?

Accepted Answer

( \frac{13}{42} ) (\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}). Here the value is (\frac{13}{42}).

Question 23

If the identity $(x+a)^2=x^2-20x+100$ is true for all $x$, what is the value of $a$?

Accepted Answer

$-10$ Expanding gives $(x+a)^2=x^2+2ax+a^2$. Comparing the coefficients of $x$ on both sides, $2a=-20$, so $a=-10$. Substituting this value also gives $a^2=100$, confirming the constant term. Exam tip: For an identity, compare coefficients of like powers of the variable.

Question 24

If k is a real number, what is the nature of the roots of the equation 25x² + 40kx + 16k² = 0?

Accepted Answer

Two equal real roots The equation can be written as 25x² + 40kx + 16k² = (5x + 4k)². Hence, (5x + 4k)² = 0 gives x = −4k/5, which is a repeated root. Equivalently, the discriminant is D = (40k)² − 4(25)(16k²) = 0, so the roots are equal and real. Option B is incorrect because distinct real roots require D > 0. Exam tip: For a quadratic equation, D = 0 indicates two equal real roots.

Question 25

If one root of (x^2+mx+49=0) is the reciprocal of the other, what can be said about (m)?

Accepted Answer

It is not possible If one root is the reciprocal of the other, the product of roots must be (1). Here the product is (49), so it is not possible.

Introduction to Quadratic Equations

Practice questions

01 What is the standard form with integer coefficients of (\frac{3x^2+2}{4}-\frac{x-5}{3}=6)?

02 If \(x=-5\) is a root of the equation \(3x^2+px-20=0\), what is the value of \(p\)?

03 Which monic quadratic equation has sum of roots (-13) and product (42)?

04 If the two roots of the quadratic equation \(x^2-2mx+49=0\) are equal, what are the possible values of \(m\)?

05 What condition on \(p\) is necessary and sufficient for the equation \(x^2+2px+36=0\) to have real roots?

06 What is the sum of reciprocals of the roots of (5x^2-17x+6=0)?

07 If 7 and 8 are the roots of the equation \(x^2-sx+p=0\), what is the value of \(s+p\)?

08 If the roots of the equation \(x^2-16x+k=0\) are real and distinct, what is the correct condition on \(k\)?

09 Which quadratic equation has a discriminant of −35?

10 What is the sum of squares of the roots of (4x^2+8x+3=0)?

11 If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2-14x+48=0\), what is the value of \((\alpha-6)(\beta-6)\)?

12 What is the standard form of the equation (x+4)(x+7)+(x−3)(x+6)=88?

13 If the roots of a monic quadratic equation are (2r) and (5r), and their sum is (21), what will be the constant term?

14 Which statement correctly describes the nature of the roots of \(x^2-10x+29=0\)?

15 Which of the following becomes a quadratic equation in the variable \(x\) after simplification?

16 What is the difference between the larger and smaller roots of \(5x^2-22x+24=0\)?

17 If the quadratic equation \(x^2-18x+c=0\) can be written as a perfect square, what is the value of \(c\)?

18 If the two roots of the equation \(x^2+bx+100=0\) are equal and each root is \(-10\), what is the value of \(b\)?

19 In which option will the sum of roots be negative and the product of roots be negative?

20 A rectangle has length \(x+9\) units and breadth \(x-6\) units. If its area is 130 square units, which quadratic equation is formed?

21 Two numbers have sum (17) and product (72). They can be roots of which quadratic equation?

22 If (\alpha,\beta) are roots of (x^2-13x+42=0), what is (\frac{1}{\alpha}+\frac{1}{\beta})?

23 If the identity \\((x+a)^2=x^2-20x+100\\) is true for all \\(x\\), what is the value of \\(a\\)?

24 If k is a real number, what is the nature of the roots of the equation 25x² + 40kx + 16k² = 0?

25 If one root of (x^2+mx+49=0) is the reciprocal of the other, what can be said about (m)?

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