Q: Which monic quadratic equation has sum of roots (-9) and product (20)?

(x^2+9x+20=0) A monic equation is (x^2-(\text{sum})x+\text{product}=0). Substituting sum (-9) gives (x^2+9x+20=0).

Q: If the roots of the quadratic equation $x^2-2kx+25=0$ are equal, what are the possible values of $k$?

$k=\pm5$ For equal roots, the discriminant must be zero. Here, $a=1$, $b=-2k$, and $c=25$, so $D=b^2-4ac=(-2k)^2-4(1)(25)=4k^2-100$. Setting $D=0$ gives $4k^2-100=0$, hence $k^2=25$ and $k=\pm5$. Therefore, option A is correct. Choosing only $k=5$ or only $k=-5$ is incomplete because both values are possible. Exam tip: For equal roots, immediately use $D=0$.

Q: What condition on $k$ is necessary for the equation $x^2+2kx+16=0$ to have real roots?

$k\leq -4$ or $k\geq 4$ For a quadratic equation $ax^2+bx+c=0$ to have real roots, its discriminant must satisfy $D=b^2-4ac\geq0$. Here, $a=1, b=2k, c=16$, so $D=(2k)^2-4(1)(16)=4k^2-64$. Thus, $4k^2-64\geq0\Rightarrow k^2\\geq16$, which gives $k\leq-4$ or $k\geq4$. In option B, $k$ lies between $-4$ and $4$, making the discriminant negative. Exam tip: Whenever real roots are asked for, begin with the condition $D\geq0$.

Q: If 4 and 5 are the roots of the equation $x^2-sx+p=0$, what is the value of $s+p$?

29 For the quadratic equation $x^2-sx+p=0$, the sum of the roots is $s$, and their product is $p$. Thus, $s=4+5=9$ and $p=4\times5=20$. Therefore, $s+p=9+20=29$, so option A is correct. Exam tip: compare the equation directly with the standard form $x^2-(\text{sum of roots})x+(\text{product of roots})=0$.

Q: If the roots of the equation $x^2-8x+k=0$ are real and distinct, what is the correct condition on $k$?

$k 0$, which gives $k 0$ specifically for distinct real roots.

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

$ \frac{40}{9} $ If the roots are $\alpha,\beta$, then $\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta$. Here $\left(-\frac{8}{3}\right)^2-2\cdot\frac{4}{3}=\frac{40}{9}$.

Q: If $\alpha$ and $\beta$ are the roots of the equation $x^2+7x+10=0$, what is the value of $(\alpha+2)(\beta+2)$?

0 By Vieta’s formulas, $\alpha+\beta=-\frac{7}{1}=-7$ and $\alpha\beta=\frac{10}{1}=10$. Therefore, $(\alpha+2)(\beta+2)=\alpha\beta+2(\alpha+\beta)+4=10+2(-7)+4=0$. Option B is only the value of $\alpha\beta$, not of the complete expression. Exam tip: For expressions involving roots, first find the sum and product of the roots using Vieta’s formulas.

Q: What is the standard form ax^2+bx+c=0) of the equation (x+2)(x+5)+(x-1)(x+4)=609?

2x^2+10x-54=0) Expand the two products: (x+2)(x+5)=x^2+7x+10) and (x-1)(x+4)=x^2+3x-4). Their sum is 2x^2+10x+6). Bringing 60 to the left gives 2x^2+10x+6-60=0), which simplifies to 2x^2+10x-54=0). Option B stops before subtracting 60, while option C does not represent the correctly simplified equation. Exam tip: In standard form, collect all terms on one side and write zero on the other side.

Q: Which of the following statements is correct for the equation \$x^2-6x+11=0\$?

It has no real roots For this quadratic equation, $a=1$, $b=-6$, and $c=11$. Its discriminant is $D=b^2-4ac=(-6)^2-4(1)(11)=36-44=-8$. Since $D 0$ gives two distinct real roots.

Question 1

Which monic quadratic equation has sum of roots (-9) and product (20)?

Accepted Answer

(x^2+9x+20=0) A monic equation is (x^2-(	ext{sum})x+	ext{product}=0). Substituting sum (-9) gives (x^2+9x+20=0).

Question 2

If the roots of the quadratic equation $x^2-2kx+25=0$ are equal, what are the possible values of $k$?

Accepted Answer

$k=\pm5$ For equal roots, the discriminant must be zero. Here, $a=1$, $b=-2k$, and $c=25$, so $D=b^2-4ac=(-2k)^2-4(1)(25)=4k^2-100$. Setting $D=0$ gives $4k^2-100=0$, hence $k^2=25$ and $k=\pm5$. Therefore, option A is correct. Choosing only $k=5$ or only $k=-5$ is incomplete because both values are possible. Exam tip: For equal roots, immediately use $D=0$.

Question 3

What condition on $k$ is necessary for the equation $x^2+2kx+16=0$ to have real roots?

Accepted Answer

$k\leq -4$ or $k\geq 4$ For a quadratic equation $ax^2+bx+c=0$ to have real roots, its discriminant must satisfy $D=b^2-4ac\geq0$. Here, $a=1, b=2k, c=16$, so $D=(2k)^2-4(1)(16)=4k^2-64$. Thus, $4k^2-64\geq0\Rightarrow k^2\geq16$, which gives $k\leq-4$ or $k\geq4$. In option B, $k$ lies between $-4$ and $4$, making the discriminant negative. Exam tip: Whenever real roots are asked for, begin with the condition $D\geq0$.

Question 4

What is the sum of reciprocals of the roots of (3x^2-10x+7=0)?

Accepted Answer

( \frac{10}{7} ) The sum of reciprocals is (\frac{\alpha+\beta}{\alpha\beta}). Here it is (\frac{\frac{10}{3}}{\frac{7}{3}}=\frac{10}{7}).

Question 5

If 4 and 5 are the roots of the equation $x^2-sx+p=0$, what is the value of $s+p$?

Accepted Answer

29 For the quadratic equation $x^2-sx+p=0$, the sum of the roots is $s$, and their product is $p$. Thus, $s=4+5=9$ and $p=4\times5=20$. Therefore, $s+p=9+20=29$, so option A is correct. Exam tip: compare the equation directly with the standard form $x^2-(\text{sum of roots})x+(\text{product of roots})=0$.

Question 6

If the roots of the equation $x^2-8x+k=0$ are real and distinct, what is the correct condition on $k$?

Accepted Answer

$k<16$ For a quadratic equation to have real and distinct roots, its discriminant $D=b^2-4ac$ must be positive. Here, $a=1, b=-8, c=k$, so $D=(-8)^2-4(1)(k)=64-4k$. Thus, $64-4k>0$, which gives $k<16$. When $k=16$, the roots are equal, so option B is incorrect. Exam tip: use $D>0$ specifically for distinct real roots.

Question 7

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

Accepted Answer

$ \frac{40}{9} $ If the roots are $\alpha,\beta$, then $\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta$. Here $\left(-\frac{8}{3}\right)^2-2\cdot\frac{4}{3}=\frac{40}{9}$.

Question 8

If $\alpha$ and $\beta$ are the roots of the equation $x^2+7x+10=0$, what is the value of $(\alpha+2)(\beta+2)$?

Accepted Answer

0 By Vieta’s formulas, $\alpha+\beta=-\frac{7}{1}=-7$ and $\alpha\beta=\frac{10}{1}=10$. Therefore, $(\alpha+2)(\beta+2)=\alpha\beta+2(\alpha+\beta)+4=10+2(-7)+4=0$. Option B is only the value of $\alpha\beta$, not of the complete expression. Exam tip: For expressions involving roots, first find the sum and product of the roots using Vieta’s formulas.

Question 9

What is the standard form ax^2+bx+c=0) of the equation (x+2)(x+5)+(x-1)(x+4)=609?

Accepted Answer

2x^2+10x-54=0) Expand the two products: (x+2)(x+5)=x^2+7x+10) and (x-1)(x+4)=x^2+3x-4). Their sum is 2x^2+10x+6). Bringing 60 to the left gives 2x^2+10x+6-60=0), which simplifies to 2x^2+10x-54=0). Option B stops before subtracting 60, while option C does not represent the correctly simplified equation. Exam tip: In standard form, collect all terms on one side and write zero on the other side.

Question 10

Which of the following statements is correct for the equation $x^2-6x+11=0$?

Accepted Answer

It has no real roots For this quadratic equation, $a=1$, $b=-6$, and $c=11$. Its discriminant is $D=b^2-4ac=(-6)^2-4(1)(11)=36-44=-8$. Since $D<0$, the equation has no real roots. Exam tip: $D=0$ gives equal real roots, while $D>0$ gives two distinct real roots.

Question 11

If one root of the equation $x^2-(m+4)x+4m=0$ is $4$, what is the other root?

Accepted Answer

m For a quadratic equation $ax^2+bx+c=0$, the product of its roots is $\frac{c}{a}$. Here, the product of the roots is $4m$. Since one root is $4$, the other root is $\frac{4m}{4}=m$. Remember that $m+4$ is the sum of the roots, not the other root.

Question 12

What is the difference between the roots of the equation $2x^2-9x+10=0$, taking the larger root minus the smaller root?

Accepted Answer

$\frac{1}{2}$ Here, the discriminant is $D=b^2-4ac=(-9)^2-4(2)(10)=81-80=1$. The difference between the roots is $\frac{\sqrt{D}}{|a|}$, so it equals $\frac{\sqrt{1}}{2}=\frac{1}{2}$. Alternatively, $2x^2-9x+10=(2x-5)(x-2)$, giving roots $\frac{5}{2}$ and $2$, whose difference is $\frac{1}{2}$. Exam tip: use $\frac{\sqrt{D}}{|a|}$ when only the difference between roots is required.

Question 13

If $x^2-14x+c=0$ is a perfect-square quadratic equation, what is the value of $c$?

Accepted Answer

49 For the quadratic to be a perfect square, its left-hand polynomial must be expressible as $x^2-14x+c=(x-7)^2$. Since $(x-7)^2=x^2-14x+49$, we get $c=49$. Exam tip: for $x^2+bx+c$ to be a perfect square, $c=\left(\frac{b}{2}\right)^2$; here, $b=-14$.

Question 14

If the two roots of the quadratic equation $x^2+bx+49=0$ are equal and each root is $-7$, what is the value of $b$?

Accepted Answer

14 The sum of the roots is $(-7)+(-7)=-14$. For a quadratic equation $x^2+bx+c=0$, the sum of the roots is $-b$. Thus, $-b=-14$, giving $b=14$. Option B is the sum of the roots, not the value of the coefficient $b$. Exam tip: Use the relation $\text{sum of roots}=-\frac{\text{coefficient of }x}{\text{coefficient of }x^2}$.

Question 15

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

Accepted Answer

(x^2+7x+12=0) In the first option, the sum is (-\frac{b}{a}=-7) and the product is (\frac{c}{a}=12). So the sum is negative and the product is positive.

Question 16

Which of the following is a quadratic equation in one variable?

Accepted Answer

$x^2-5x+6=0$ A quadratic equation has the form $ax^2+bx+c=0$, where $a\ne0$. In option A, the highest power of x is 2. Option B is cubic, while D is linear. Exam tip: check the degree after simplifying fractions.

Question 17

Two numbers have sum (13) and product (40). They can be roots of which quadratic equation?

Accepted Answer

(x^2-13x+40=0) If the sum of roots is (13) and product is (40), the equation is (x^2-13x+40=0). Remember the monic form formula.

Question 18

If (\alpha,\beta) are roots of (x^2-7x+12=0), what is (\frac{1}{\alpha}+\frac{1}{\beta})?

Accepted Answer

( \frac{7}{12} ) (\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}). Here the value is (\frac{7}{12}).

Question 19

Which of the following quadratic equations will have two equal and negative roots?

Accepted Answer

x² + 16x + 64 = 0 For option A, $x^2+16x+64=(x+8)^2$. Thus, $(x+8)^2=0$ gives both roots as $x=-8$, so they are equal and negative. In option B, $(x-8)^2=0$, giving equal but positive roots. Exam tip: For equal roots, first check that the discriminant is zero, then determine the sign of the root.

Question 20

If $x+a$^2 = x^2 - 12x + 36 holds for all real x, what is the value of a?

Accepted Answer

-6 Expanding gives $x+a$^2 = x^2 + 2ax + a^2\). Comparing the coefficients of x, $2a=-12$, so $a=-6$. The constant term gives $a^2=36$, which alone allows both 6 and -6; the coefficient of x identifies -6 uniquely. Exam tip: For an identity, compare coefficients of like powers of the variable.

Question 21

If $k$ is a real constant, what is the nature of the roots of the equation $9x^2+12kx+4k^2=0$?

Accepted Answer

Two equal real roots The equation can be rewritten as $(3x+2k)^2=0$. Thus $3x+2k=0$, giving the repeated root $x=-\frac{2k}{3}$. Equivalently, the discriminant is $D=(12k)^2-4(9)(4k^2)=0$, so the roots are equal and real. Exam tip: for a quadratic equation, $D=0$ indicates two equal real roots.

Question 22

If one root of (x^2+mx+25=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 (25), so it is not possible.

Question 23

The roots of (x^2+px+16=0) are equal and positive. What is the value of (p)?

Accepted Answer

(-8) For equal roots, (p^2-64=0) gives (p=\pm8). For equal positive roots, (-\frac{p}{2}>0), so (p=-8).

Question 24

If (x=-1) is not a root of (kx^2+3x+2=0), what condition must (k) satisfy?

Accepted Answer

(k
eq1) The direct answer is option A: k ≠ 1. A number is a root of an equation when substituting that number makes the left-hand side equal to zero. Put x = -1 into kx² + 3x + 2. First, (-1)² = 1, so kx² = k. Next, 3x = 3(-1) = -3. Therefore the expression becomes k - 3 + 2 = k - 1. Since -1 is not a root, this value must not be zero: k - 1 ≠ 0, hence k ≠ 1. Option A is correct because it states exactly this condition. Option B, k = 1, is wrong because then the value is 1 - 1 = 0, so -1 would actually be a root. Option C, k ≠ 3, is not the condition obtained; for example, k = 1 satisfies k ≠ 3 but still makes -1 a root. Option D, k = 3, is also unnecessarily specific and does not describe all allowed values; k = 3 is allowed, but many other values are allowed too. Exam cue: for a proposed root, substitute it and require the expression to be zero; for a non-root, require it to be nonzero.

Question 25

Which quadratic equation has sum of roots (0) and product (-81)?

Accepted Answer

(x^2-81=0) The monic equation is (x^2-(	ext{sum})x+	ext{product}=0). Using sum (0) and product (-81) gives (x^2-81=0).

Introduction to Quadratic Equations

Practice questions

01 Which monic quadratic equation has sum of roots (-9) and product (20)?

02 If the roots of the quadratic equation \(x^2-2kx+25=0\) are equal, what are the possible values of \(k\)?

03 What condition on \(k\) is necessary for the equation \(x^2+2kx+16=0\) to have real roots?

04 What is the sum of reciprocals of the roots of (3x^2-10x+7=0)?

05 If 4 and 5 are the roots of the equation \(x^2-sx+p=0\), what is the value of \(s+p\)?

06 If the roots of the equation \(x^2-8x+k=0\) are real and distinct, what is the correct condition on \(k\)?

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

08 If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2+7x+10=0\), what is the value of \((\alpha+2)(\beta+2)\)?

09 What is the standard form ax^2+bx+c=0) of the equation (x+2)(x+5)+(x-1)(x+4)=609?

10 Which of the following statements is correct for the equation \\(x^2-6x+11=0\\)?

11 If one root of the equation \(x^2-(m+4)x+4m=0\) is \(4\), what is the other root?

12 What is the difference between the roots of the equation \(2x^2-9x+10=0\), taking the larger root minus the smaller root?

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

14 If the two roots of the quadratic equation \(x^2+bx+49=0\) are equal and each root is \(-7\), what is the value of \(b\)?

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

16 Which of the following is a quadratic equation in one variable?

17 Two numbers have sum (13) and product (40). They can be roots of which quadratic equation?

18 If (\alpha,\beta) are roots of (x^2-7x+12=0), what is (\frac{1}{\alpha}+\frac{1}{\beta})?

19 Which of the following quadratic equations will have two equal and negative roots?

20 If \\(x+a\\)^2 = x^2 - 12x + 36 holds for all real x, what is the value of a?

21 If \(k\) is a real constant, what is the nature of the roots of the equation \(9x^2+12kx+4k^2=0\)?

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

23 The roots of (x^2+px+16=0) are equal and positive. What is the value of (p)?

24 If (x=-1) is not a root of (kx^2+3x+2=0), what condition must (k) satisfy?

25 Which quadratic equation has sum of roots (0) and product (-81)?

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