Q: If the roots of $x^2-2mx+64=0$ are equal, what are the possible values of $m$?

m = \pm 8 For equal roots the discriminant must be zero: $D=0$. In the given equation $a=1,\ b=-2m,\ c=64$. So $D=b^2-4ac=(-2m)^2-4\cdot1\cdot64=4m^2-256$. Setting $D=0$ gives $4m^2-256=0\Rightarrow m^2=64\Rightarrow m=\pm8$. Thus $m=\pm8$ is correct. The option $m=\pm16$ is incorrect because it yields a nonzero discriminant (roots are not equal). Exam tip: always compute $b^2-4ac$ first for equal/distinct roots and solve the resulting equation for the parameter.

Q: What is the condition on p for the equation $x^2+2px+49=0$ to have real roots?

$p\leq -7$ or $p\geq 7$ For real roots the discriminant must satisfy $D\ge0$. Here $D=(2p)^2-4\cdot1\cdot49=4p^2-196=4(p^2-49)$. Requiring $D\ge0$ gives $p^2-49\ge0$ or $p^2\ge49$, hence $p\le-7$ or $p\ge7$. Option B is incorrect because for $-7<p<7$ we have $p^2<49$ so $D<0$; option C is just a special value and not the general condition; option D is too broad and includes values that do not give real roots. Exam tip: For parameter problems, compute $D=b^2-4ac$ first and set $D\ge0$ to find the range of the parameter.

Q: What is the sum of reciprocals of the roots of (6x^2-19x+10=0)?

( \frac{19}{10} ) The sum of reciprocals is (\frac{\alpha+\beta}{\alpha\beta}). Here it is (\frac{\frac{19}{6}}{\frac{10}{6}}=\frac{19}{10}).

Q: If 8 and 9 are the roots of the equation $x^2 - s x + p = 0$, what is the value of $s+p$?

89 By Vieta’s relations for $x^2+bx+c=0$, the sum of roots = $-b$ and the product = $c$. For $x^2 - s x + p = 0$, the sum of the roots equals $s$ and the product equals $p$. With roots 8 and 9 we get $s=8+9=17$ and $p=8\times9=72$. Hence $s+p=17+72=89$. Note that 17 is only the sum (s) and 72 is only the product (p), so they are incorrect as s+p. Exam tip: apply Vieta’s formulas directly instead of recomputing coefficients each time.

Q: If the roots of $x^2-20x+k=0$ are real and distinct, what is the correct condition on $k$?

$k 0$, hence $400-4k>0$ which gives $k 100$ the roots are complex. Exam tip: factor common multiples ($D=4(100-k)$) to check the sign quickly.

Q: Which of the following quadratic equations has discriminant -47?

x^2+x+12=0 Discriminant is defined by $D=b^2-4ac$. For option A, $a=1,\;b=1,\;c=12$, so $D=1^2-4\cdot1\cdot12=1-48=-47$, which matches. The closest distractor, option D, gives $D=(-8)^2-4\cdot1\cdot30=64-120=-56$, not -47. Exam tip: always identify $a,b,c$ first and compute $b^2-4ac$ carefully, watching signs of b and c.

Q: What is the sum of squares of the roots of $6x^2+7x+2=0$?

$ \frac{25}{36} $ $\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta$. Here $\left(-\frac{7}{6}\right)^2-2\cdot\frac{1}{3}=\frac{25}{36}$.

Q: If $\alpha$ and $\beta$ are the roots of $x^2-18x+80=0$, what is the value of $(\alpha-8)(\beta-8)$?

0 Use the identity $(\alpha-8)(\beta-8)=\alpha\beta-8(\alpha+\beta)+64$. For $x^2-18x+80=0$, by Vieta $\alpha+\beta=18$ and $\alpha\beta=80$. Therefore the value is $80-8\times18+64=80-144+64=0$. A common mistake is to pick $80$ (the product $\alpha\beta$) and ignore the linear shift terms. Exam tip: Apply Vieta's formulas directly or set $y=x-8$ to transform the expression quickly.

Question 1

If the roots of $x^2-2mx+64=0$ are equal, what are the possible values of $m$?

Accepted Answer

m = \pm 8 For equal roots the discriminant must be zero: $D=0$. In the given equation $a=1,\ b=-2m,\ c=64$. So $D=b^2-4ac=(-2m)^2-4\cdot1\cdot64=4m^2-256$. Setting $D=0$ gives $4m^2-256=0\Rightarrow m^2=64\Rightarrow m=\pm8$. Thus $m=\pm8$ is correct. The option $m=\pm16$ is incorrect because it yields a nonzero discriminant (roots are not equal). Exam tip: always compute $b^2-4ac$ first for equal/distinct roots and solve the resulting equation for the parameter.

Question 2

What is the condition on p for the equation $x^2+2px+49=0$ to have real roots?

Accepted Answer

$p\leq -7$ or $p\geq 7$ For real roots the discriminant must satisfy $D\ge0$. Here $D=(2p)^2-4\cdot1\cdot49=4p^2-196=4(p^2-49)$. Requiring $D\ge0$ gives $p^2-49\ge0$ or $p^2\ge49$, hence $p\le-7$ or $p\ge7$. Option B is incorrect because for $-7<p<7$ we have $p^2<49$ so $D<0$; option C is just a special value and not the general condition; option D is too broad and includes values that do not give real roots. Exam tip: For parameter problems, compute $D=b^2-4ac$ first and set $D\ge0$ to find the range of the parameter.

Question 3

What is the sum of reciprocals of the roots of (6x^2-19x+10=0)?

Accepted Answer

( \frac{19}{10} ) The sum of reciprocals is (\frac{\alpha+\beta}{\alpha\beta}). Here it is (\frac{\frac{19}{6}}{\frac{10}{6}}=\frac{19}{10}).

Question 4

If 8 and 9 are the roots of the equation $x^2 - s x + p = 0$, what is the value of $s+p$?

Accepted Answer

89 By Vieta’s relations for $x^2+bx+c=0$, the sum of roots = $-b$ and the product = $c$. For $x^2 - s x + p = 0$, the sum of the roots equals $s$ and the product equals $p$. With roots 8 and 9 we get $s=8+9=17$ and $p=8\times9=72$. Hence $s+p=17+72=89$. Note that 17 is only the sum (s) and 72 is only the product (p), so they are incorrect as s+p. Exam tip: apply Vieta’s formulas directly instead of recomputing coefficients each time.

Question 5

If the roots of $x^2-20x+k=0$ are real and distinct, what is the correct condition on $k$?

Accepted Answer

$k<100$ For a quadratic to have real and distinct roots the discriminant $D=b^2-4ac$ must be positive. Here $a=1,\;b=-20,\;c=k$, so $D=(-20)^2-4\cdot1\cdot k=400-4k$. Real and distinct roots require $D>0$, hence $400-4k>0$ which gives $k<100$. If $k=100$ the roots are equal, and if $k>100$ the roots are complex. Exam tip: factor common multiples ($D=4(100-k)$) to check the sign quickly.

Question 6

Which of the following quadratic equations has discriminant -47?

Accepted Answer

x^2+x+12=0 Discriminant is defined by $D=b^2-4ac$. For option A, $a=1,\;b=1,\;c=12$, so $D=1^2-4\cdot1\cdot12=1-48=-47$, which matches. The closest distractor, option D, gives $D=(-8)^2-4\cdot1\cdot30=64-120=-56$, not -47. Exam tip: always identify $a,b,c$ first and compute $b^2-4ac$ carefully, watching signs of b and c.

Question 7

What is the sum of squares of the roots of $6x^2+7x+2=0$?

Accepted Answer

$ \frac{25}{36} $ $\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta$. Here $\left(-\frac{7}{6}\right)^2-2\cdot\frac{1}{3}=\frac{25}{36}$.

Question 8

If $\alpha$ and $\beta$ are the roots of $x^2-18x+80=0$, what is the value of $(\alpha-8)(\beta-8)$?

Accepted Answer

0 Use the identity $(\alpha-8)(\beta-8)=\alpha\beta-8(\alpha+\beta)+64$. For $x^2-18x+80=0$, by Vieta $\alpha+\beta=18$ and $\alpha\beta=80$. Therefore the value is $80-8\times18+64=80-144+64=0$. A common mistake is to pick $80$ (the product $\alpha\beta$) and ignore the linear shift terms. Exam tip: Apply Vieta's formulas directly or set $y=x-8$ to transform the expression quickly.

Question 9

What is the standard form of ((x+5)(x+8)+(x-4)(x+7)=110)?

Accepted Answer

(2x^2+16x-98=0) The standard form of a quadratic equation is written as a polynomial equal to zero. First expand each product carefully. We get $x+5)(x+8)=x^2+13x+40$ and $x-4)(x+7)=x^2+3x-28$. Adding them gives $2x^2+16x+12$, so the left side is a quadratic expression.

The original equation says this expression equals 110. Move 110 to the left by subtracting it from both sides: $2x^2+16x+12-110=0$. Combining the constants gives $2x^2+16x-98=0$. Therefore, option A is correct. Option C loses a factor of 2, while option D does not correctly expand the products or rearrange the constant.

Question 10

If the roots of a monic quadratic equation are (3r) and (4r), and their sum is (28), what will be the constant term?

Accepted Answer

(192) From (3r+4r=28), we get (r=4), so the roots are (12) and (16). The constant term is the product of roots (192).

Question 11

Which of the following statements is true for the equation $x^2 - 12x + 40 = 0$?

Accepted Answer

It has no real roots Compute the discriminant: $D=b^2-4ac = (-12)^2 - 4\cdot1\cdot40 = 144 - 160 = -16 < 0$. When $D<0$ a quadratic has no real roots (the roots are non-real complex conjugates). Thus option A is correct. Option B is wrong because equal (repeated) real roots occur only if $D=0$. Option C is wrong because two distinct real roots require $D>0$. Option D is wrong because the product of the roots equals $c/a = 40$, not $-40$. Exam tip: always evaluate $D$ first to decide the nature of roots; use sum = $-b/a$ and product = $c/a$ for quick checks.

Question 12

If one root of the quadratic equation $x^2-(m+7)x+7m=0$ is 7, what is the other root?

Accepted Answer

$m$ For a quadratic $ax^2+bx+c=0$, the product of roots equals $c/a$. Here $a=1$ and $c=7m$, so the product is $7m$. Given one root is 7, the other root is $\dfrac{7m}{7}=m$. Check with the sum: sum = $-b/a$ = $m+7$, which matches $7+m$. The closest distractor (option B: $7m$) confuses the product with a root. Exam tip: always apply sum = $-b/a$ and product = $c/a$, and watch the signs of coefficients.

Question 13

What is the difference between the roots of the equation $7x^{2}-30x+32=0$?

Accepted Answer

$\tfrac{2}{7}$ Compute the discriminant: for $7x^{2}-30x+32=0$, $a=7,\;b=-30,\;c=32$ so $D=b^{2}-4ac=900-896=4$. The difference between roots equals $|r_{1}-r_{2}|=\dfrac{\sqrt{D}}{a}$. Therefore the difference is $\dfrac{\sqrt{4}}{7}=\dfrac{2}{7}$. Note: the option $\tfrac{4}{7}$ results from mistakenly using $D/a$, and $\tfrac{16}{7}$ is one of the roots (the other root is 2), not their difference. Exam tip: use $|r_{1}-r_{2}|=\sqrt{D}/|a|$ to get the answer quickly without finding both roots explicitly.

Question 14

If $x^2+24x+c=0$ is a perfect-square quadratic, what is the value of $c$?

Accepted Answer

144 A perfect-square trinomial must match $(x+b)^2 = x^2 + 2bx + b^2$. Here the coefficient of $x$ is 24, so $2b=24$ giving $b=12$, and hence $c=b^2=12^2=144$. Alternatively use discriminant: $b^2-4ac=24^2-4\cdot1\cdot c=576-4c=0$ which yields $c=144$. Note: 576 might look tempting because it is $24^2$, but the constant term must be $b^2$ with $2b=24$, not $24^2$. Exam tip: either complete the square or set the discriminant to zero to solve such questions quickly.

Question 15

If both roots of $x^2+bx+144=0$ are equal and each is $-12$, what is the value of $b$?

Accepted Answer

24 For a quadratic, the sum of roots is $\alpha+\beta=-b/a$. Here $a=1$ and both roots are $-12$, so the sum is $-24$. Thus $-b=-24$ which gives $b=24$. Alternatively, for a repeated root use $-b/(2a)$: $-b/2=-12$ leads to the same result. The distractor $-24$ is the sign-error mirror of the correct sum, so it is incorrect. Exam tip: use $\alpha+\beta=-b/a$ and $\alpha\beta=c/a$ and double-check signs when substituting.

Question 16

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

Accepted Answer

(x^2-9x+20=0) The direct answer is option A: x² - 9x + 20 = 0. For ax² + bx + c = 0, the sum of roots is -b/a and the product is c/a. Since every equation here is monic, a = 1. In option A, b = -9 and c = 20, so the sum is -(-9)/1 = 9, positive, and the product is 20/1 = 20, also positive. Therefore A satisfies both requirements. In option B, the sum is -(9) = -9, which is negative, although the product 20 is positive; it fails the first condition. In option C, the sum is 9, positive, but the product is -20, negative; it fails the second condition. In option D, the sum is -9 and the product is -20, so both signs fail. Thus only A works. Another check is factoring A as (x - 4)(x - 5) = 0; its roots are 4 and 5, whose sum and product are both positive. Memory cue: in a monic quadratic, the sign before x is opposite to the root sum, while the constant sign gives the product sign.

Question 17

A rectangle has length $(x+10)$ and breadth $(x-7)$. If its area is $160$, which equation is correct?

Accepted Answer

$x^2+3x-230=0$ Area gives $(x+10)(x-7)=160$. Expanding the left side yields $x^2+3x-70=160$. Bringing 160 to the left gives $x^2+3x-70-160=0$, i.e. $x^2+3x-230=0$, so option A is correct. The closest wrong option is B, which has the constant term $-160$ instead of $-230$ — a typical mistake from not moving the right-hand term across correctly. Exam tip: first expand, then move all terms to one side to form the quadratic and carefully check signs and arithmetic.

Question 18

Two numbers have sum (19) and product (90). They can be roots of which quadratic equation?

Accepted Answer

(x^2-19x+90=0) If the sum of roots is (19) and product is (90), the equation is (x^2-19x+90=0). Remember the monic form formula.

Question 19

If (\alpha,\beta) are roots of (x^2-17x+72=0), what is (\frac{1}{\alpha}+\frac{1}{\beta})?

Accepted Answer

( \frac{17}{72} ) (\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}). Here the value is (\frac{17}{72}).

Question 20

If $(x+a)^2 = x^2 + 28x + 196$, what is the value of $a$?

Accepted Answer

14 Using the identity $(x+a)^2 = x^2 + 2ax + a^2$, compare coefficients with $x^2 + 28x + 196$. From the coefficient of $x$ we get $2a = 28$, so $a = 14$. Also $a^2 = 196$ holds for $a=14$. The option $-14$ is incorrect because it would give $2a = -28$, which contradicts the given linear coefficient. Exam tip: equate coefficients of like powers of $x$ to find unknown parameters quickly.

Question 21

What is the nature of the roots of the quadratic equation $36x^2-60kx+25k^2=0$?

Accepted Answer

Two equal real roots The quadratic is a perfect square: $(6x-5k)^2=0$. Alternatively compute the discriminant: $\Delta=b^2-4ac=(-60k)^2-4\cdot36\cdot25k^2=3600k^2-3600k^2=0$. A zero discriminant means the two roots are equal and real (root $x=5k/6$). Option B is wrong because that requires $\Delta>0$; option C is wrong because that requires $\Delta<0$; option D is wrong because the nature does not depend on a special value of k for real k — it is always equal roots. Exam tip: check the discriminant first or try factoring into a perfect square to save time.

Question 22

If one root of (x^2+mx+64=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 (64), so it is not possible.

Question 23

The roots of (x^2+px+49=0) are equal and negative. What is the value of (p)?

Accepted Answer

(14) For equal roots, (p^2-196=0) gives (p=\pm14). For equal negative roots, (-\frac{p}{2}<0), so (p=14).

Question 24

If (x=3) is not a root of (kx^2-8x+5=0), what condition must (k) satisfy?

Accepted Answer

(k
eq \frac{19}{9}) Putting (x=3), the left side becomes (9k-24+5=9k-19). For it not to be a root, (9k-19
eq0), so (k
eq\frac{19}{9}).

Question 25

Which quadratic equation has sum of roots (0) and product (-169)?

Accepted Answer

(x^2-169=0) The monic equation is (x^2-(	ext{sum})x+	ext{product}=0). Using sum (0) and product (-169) gives (x^2-169=0).

Introduction to Quadratic Equations

Practice questions

01 If the roots of \(x^2-2mx+64=0\) are equal, what are the possible values of \(m\)?

02 What is the condition on p for the equation \(x^2+2px+49=0\) to have real roots?

03 What is the sum of reciprocals of the roots of (6x^2-19x+10=0)?

04 If 8 and 9 are the roots of the equation \(x^2 - s x + p = 0\), what is the value of \(s+p\)?

05 If the roots of \(x^2-20x+k=0\) are real and distinct, what is the correct condition on \(k\)?

06 Which of the following quadratic equations has discriminant -47?

07 What is the sum of squares of the roots of \(6x^2+7x+2=0\)?

08 If \(\alpha\) and \(\beta\) are the roots of \(x^2-18x+80=0\), what is the value of \((\alpha-8)(\beta-8)\)?

09 What is the standard form of ((x+5)(x+8)+(x-4)(x+7)=110)?

10 If the roots of a monic quadratic equation are (3r) and (4r), and their sum is (28), what will be the constant term?

11 Which of the following statements is true for the equation \(x^2 - 12x + 40 = 0\)?

12 If one root of the quadratic equation \(x^2-(m+7)x+7m=0\) is 7, what is the other root?

13 What is the difference between the roots of the equation \(7x^{2}-30x+32=0\)?

14 If \(x^2+24x+c=0\) is a perfect-square quadratic, what is the value of \(c\)?

15 If both roots of \(x^2+bx+144=0\) are equal and each is \(-12\), what is the value of \(b\)?

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

17 A rectangle has length \((x+10)\) and breadth \((x-7)\). If its area is \(160\), which equation is correct?

18 Two numbers have sum (19) and product (90). They can be roots of which quadratic equation?

19 If (\alpha,\beta) are roots of (x^2-17x+72=0), what is (\frac{1}{\alpha}+\frac{1}{\beta})?

20 If \((x+a)^2 = x^2 + 28x + 196\), what is the value of \(a\)?

21 What is the nature of the roots of the quadratic equation \(36x^2-60kx+25k^2=0\)?

22 If one root of (x^2+mx+64=0) is the reciprocal of the other, what can be said about (m)?

23 The roots of (x^2+px+49=0) are equal and negative. What is the value of (p)?

24 If (x=3) is not a root of (kx^2-8x+5=0), what condition must (k) satisfy?

25 Which quadratic equation has sum of roots (0) and product (-169)?

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