Q: If \$x+a)^2=x^2+10x+25\$ is an identity, what is the value of \$a\$?

5 Expanding gives \$x+a)^2=x^2+2ax+a^2\$. Comparing the coefficients of \$x\$ on both sides, \$2a=10\$, so \$a=5\$. Option B is incorrect because \$a=-5\$ would make the coefficient of \$x\$ equal to \$-10\$. Exam tip: In an identity, compare the coefficients of like powers of the variable.

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

Two equal real roots The equation can be written as $4x^2-4kx+k^2=(2x-k)^2=0$. Hence $2x-k=0$, giving the repeated root $x=\frac{k}{2}$. Therefore, the roots are equal and real. Exam tip: use the discriminant $D=b^2-4ac$; here $D=0$, so the roots are equal. Thus option B is incorrect because the roots are not distinct.

Q: Which quadratic equation has sum of roots (0) and product (-49)?

(x^2-49=0) The monic equation is (x^2-(\text{sum})x+\text{product}=0). Using sum (0) and product (-49) gives (x^2-49=0).

Q: If in $2x^2+kx+18=0$, the product of roots is (-3) times the sum of roots, what is (k)?

(6) The direct answer is A: $k=6$. For $2x^2+kx+18=0$, the sum of roots is $-k/2$, and the product is $18/2=9$. The statement says product equals -3 times the sum, so $9=-3(-k/2)=3k/2$. Multiplying by 2 gives $18=3k$, hence $k=6$. Option A is correct. Option B, $-6$, makes the sum 3, so -3 times it is -9, not 9. Option C, 12, gives product relation 18, not 9. Option D, -12, gives the wrong sign and magnitude. The key is to use Vieta’s formulas with signs carefully. Memory cue: sum is $-b/a$, product is $c/a$.

Q: What is the sum of cubes of the roots of (x^2-2x-3=0)?

(26) The sum of roots is (2) and the product is (-3). (\alpha^3+\beta^3=(\alpha+\beta)^3-3\alpha\beta(\alpha+\beta)=8+18=26).

Q: If the roots of \$x^2+bx+c=0\$ are \$-2\$ and \$7\$, what is the value of \$b+c\$?

-19 For the monic quadratic equation \$x^2+bx+c=0\$, the sum of the roots is \$-b\$ and their product is \$c\$. Here, the sum is \$-2+7=5\$, so \$-b=5\$, giving \$b=-5\$. The product is \$(-2)\\times7=-14\$, so \$c=-14\$. Therefore, \$b+c=-5-14=-19\$. Exam tip: For \$ax^2+bx+c=0\$, use the root sum \$-b/a\$ and root product \$c/a\$.

Q: Which of the following quadratic equations has one root $x=0$ and the other root positive?

$x^2-6x=0$ In option A, $x^2-6x=x(x-6)$, so the roots are $x=0$ and $x=6$. Since $6$ is positive, this option is correct. Option B has roots $0$ and $-6$, so its other root is negative; option C has no real roots, and option D does not have zero as a root. A quick exam check is that an equation with $x=0$ as a root must have a zero constant term.

Question 1

Which of the following quadratic equations has two equal and positive roots?

Accepted Answer

x² − 12x + 36 = 0 In option A, x² − 12x + 36 = (x − 6)², so the repeated root is x = 6, which is positive. Option B has equal roots of −6, option C has two distinct roots, 6 and −6, and option D has no real roots. Exam tip: equal roots require a discriminant of zero, and the repeated root must be positive.

Question 2

If $x+a)^2=x^2+10x+25$ is an identity, what is the value of $a$?

Accepted Answer

5 Expanding gives $x+a)^2=x^2+2ax+a^2$. Comparing the coefficients of $x$ on both sides, $2a=10$, so $a=5$. Option B is incorrect because $a=-5$ would make the coefficient of $x$ equal to $-10$. Exam tip: In an identity, compare the coefficients of like powers of the variable.

Question 3

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

Accepted Answer

Two equal real roots The equation can be written as $4x^2-4kx+k^2=(2x-k)^2=0$. Hence $2x-k=0$, giving the repeated root $x=\frac{k}{2}$. Therefore, the roots are equal and real. Exam tip: use the discriminant $D=b^2-4ac$; here $D=0$, so the roots are equal. Thus option B is incorrect because the roots are not distinct.

Question 4

If one root of (x^2+mx+16=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 (16), so it is not possible.

Question 5

The roots of (x^2+px+9=0) are equal and negative. What is the value of (p)?

Accepted Answer

(6) For equal roots, (p^2-36=0) gives (p=\pm6). For equal negative roots, (-\frac{p}{2}<0), so (p=6).

Question 6

Which quadratic equation has sum of roots (0) and product (-49)?

Accepted Answer

(x^2-49=0) The monic equation is (x^2-(	ext{sum})x+	ext{product}=0). Using sum (0) and product (-49) gives (x^2-49=0).

Question 7

If in $2x^2+kx+18=0$, the product of roots is (-3) times the sum of roots, what is (k)?

Accepted Answer

(6) The direct answer is A: $k=6$. For $2x^2+kx+18=0$, the sum of roots is $-k/2$, and the product is $18/2=9$. The statement says product equals -3 times the sum, so $9=-3(-k/2)=3k/2$. Multiplying by 2 gives $18=3k$, hence $k=6$. Option A is correct. Option B, $-6$, makes the sum 3, so -3 times it is -9, not 9. Option C, 12, gives product relation 18, not 9. Option D, -12, gives the wrong sign and magnitude. The key is to use Vieta’s formulas with signs carefully. Memory cue: sum is $-b/a$, product is $c/a$.

Question 8

What is the sum of cubes of the roots of (x^2-2x-3=0)?

Accepted Answer

(26) The sum of roots is (2) and the product is (-3). (\alpha^3+\beta^3=(\alpha+\beta)^3-3\alpha\beta(\alpha+\beta)=8+18=26).

Question 9

If the roots of $x^2+bx+c=0$ are $-2$ and $7$, what is the value of $b+c$?

Accepted Answer

-19 For the monic quadratic equation $x^2+bx+c=0$, the sum of the roots is $-b$ and their product is $c$. Here, the sum is $-2+7=5$, so $-b=5$, giving $b=-5$. The product is $(-2)\times7=-14$, so $c=-14$. Therefore, $b+c=-5-14=-19$. Exam tip: For $ax^2+bx+c=0$, use the root sum $-b/a$ and root product $c/a$.

Question 10

Which of the following quadratic equations has one root $x=0$ and the other root positive?

Accepted Answer

$x^2-6x=0$ In option A, $x^2-6x=x(x-6)$, so the roots are $x=0$ and $x=6$. Since $6$ is positive, this option is correct. Option B has roots $0$ and $-6$, so its other root is negative; option C has no real roots, and option D does not have zero as a root. A quick exam check is that an equation with $x=0$ as a root must have a zero constant term.

Question 11

If $a=-1$ is substituted in the equation $(a+1)x^2+(a^2-1)x+2=0$, what type of result is obtained?

Accepted Answer

Contradictory statement Substituting $a=-1$ gives $a+1=0$ and $a^2-1=1-1=0$. Hence the equation becomes $0x^2+0x+2=0$, or $2=0$. This is a false, contradictory statement and has no solution. It is not a linear equation because the coefficient of $x$ is also zero. In parameter-based equations, first check the coefficients of $x^2$, $x$, and the constant term separately.

Question 12

What is the standard form of ((x-1)^2+(x-2)^2=(x-3)^2)?

Accepted Answer

(x^2-2x-4=0) Expanding gives left side (2x^2-6x+5) and right side (x^2-6x+9). Subtracting gives (x^2-4=0).

Question 13

If $x=1$ is a root of the quadratic equation $x^2+(2k-1)x+k=0$, what is the value of $k$?

Accepted Answer

0 A root must make the left-hand side of the equation equal to zero. Substituting $x=1$ gives $1+(2k-1)+k=0$, so $3k=0$ and hence $k=0$. The values $\pm\frac{1}{3}$ result from incorrect simplification. Exam tip: substitute the given root carefully and collect the parameter terms before solving.

Question 14

The roots of the equation $x^2-4x+1=0$ are $\alpha$ and $\beta$. What is the value of $\alpha+\beta+\alpha\beta$?

Accepted Answer

5 For a quadratic equation $ax^2+bx+c=0$, the sum of the roots is $-\frac{b}{a}$ and their product is $\frac{c}{a}$. Here, $a=1, b=-4, c=1$, so $\alpha+\beta=4$ and $\alpha\beta=1$. Therefore, $\alpha+\beta+\alpha\beta=4+1=5$. Exam tip: Use the sum and product of roots directly instead of solving the quadratic equation.

Question 15

If the roots of $x^2+px+q=0$ are 1 and $p$, which relation for $q$ is correct?

Accepted Answer

$q=p$ For the quadratic equation $x^2+px+q=0$, the product of the roots equals the constant term $q$. Since the roots are 1 and $p$, $q=1\cdot p=p$. The sum condition also gives $1+p=-p$, so $p=-\frac{1}{2}$; nevertheless, the required relation remains $q=p$. Exam tip: In $x^2+bx+c=0$, the product of the roots is always $c$.

Question 16

Which of the following is a quadratic equation in x but is not written in standard form?

Accepted Answer

$x(x-3)=4$ Bringing $x(x-3)=4$ to one side gives $x^2-3x-4=0$, where the coefficient of $x^2$ is 1. Hence it is quadratic. Option D is an identity because both sides are identical. Exam tip: check that the highest power is 2.

Question 17

Which equation will have a negative product of roots?

Accepted Answer

(2x^2+3x-5=0) Direct answer: Option A, $2x^2+3x-5=0$. For $ax^2+bx+c=0$, the product of roots is $c/a$. We only need the sign. Option A has a=2 and c=-5, so the product is $-5/2$, which is negative. Option B has c=5 and a=2, giving $5/2>0$, so it is not negative. Option C also has a=2 and c=5, so its product is again positive; the changed sign of the x term does not affect the product. Option D has c=0, so its product is 0, not negative. Thus A is the only correct choice. Do not confuse the sign of b with the sign of the product: the product depends on c/a.

Question 18

A right triangle has a base of $(x+1)$ units, a height of $(x+3)$ units, and an area of $30$ square units. Which quadratic equation is obtained from this condition?

Accepted Answer

$x^2+4x-57=0$ The area of a right triangle is $\frac{1}{2}\times\text{base}\times\text{height}$. Hence, $\frac{1}{2}(x+1)(x+3)=30$, giving $(x+1)(x+3)=60$. Expanding, $x^2+4x+3=60$, so the quadratic equation is $x^2+4x-57=0$. Option D results from mishandling the factor $\frac{1}{2}$. Exam tip: For triangle-area problems, include the factor $\frac{1}{2}$ before expanding the algebraic expression.

Question 19

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

Accepted Answer

4 By Vieta’s formula, the sum of the roots of $ax^2+bx+c=0$ is $\alpha+\beta=-\frac{b}{a}$. Here, $a=1$ and $b=2$, so $\alpha+\beta=-2$. Therefore, $(\alpha+\beta)^2=(-2)^2=4$. Exam tip: square the complete sum, including its sign; the negative sign disappears only after squaring.

Question 20

The equation $x^2+4x+k=0$ has no real roots. What is the correct condition on $k$?

Accepted Answer

$k>4$ A quadratic equation has no real roots when its discriminant $D=b^2-4ac$ is negative. Here, $a=1, b=4, c=k$, so $D=4^2-4(1)(k)=16-4k$. Thus, $16-4k<0$, which gives $k>4$. At $k=4$, the discriminant is zero and the equation has one repeated real root, so that option is incorrect. Exam tip: for ‘no real roots,’ always use the condition $D<0$.

Question 21

If the equation $x^2-2ax+a^2-9=0$ is given, what is the difference between the larger and smaller roots?

Accepted Answer

6 The equation can be rewritten as $(x-a)^2-9=0$. Thus, $(x-a)^2=9$, giving the roots $x=a+3$ and $x=a-3$. The difference between the larger and smaller roots is $(a+3)-(a-3)=6$. Option D represents only one root, while option C is not generally the difference between the roots. Exam tip: For this form, complete the square and identify the two roots directly.

Question 22

What is the standard quadratic form of ((6x+1)(x-4)=5x)?

Accepted Answer

(6x^2-28x-4=0) Here ((6x+1)(x-4)=6x^2-23x-4), and subtracting (5x) gives (6x^2-28x-4=0). First expand and then bring all terms to one side.

Question 23

What is the standard form of the equation $2x-3$^2+(x+5)^2=34?

Accepted Answer

$5x^2-2x=0$ Expand the squares: $ (2x-3)^2=4x^2-12x+9 $ and $ (x+5)^2=x^2+10x+25 $. This gives $5x^2-2x+34=34$; subtracting 34 from both sides yields $5x^2-2x=0$. This matches the standard quadratic form $ax^2+bx+c=0$, with $c=0$. Exam tip: after expansion, bring all terms to one side and equate the result to zero.

Question 24

Write (\frac{x^2+1}{3}-\frac{x-2}{2}=5) in standard form with integer coefficients.

Accepted Answer

(2x^2-3x-22=0) Multiplying the whole equation by (6) gives (2x^2+2-3x+6=30). Therefore the standard form is (2x^2-3x-22=0).

Question 25

If x = -4 is a root of the equation 2x² + px − 12 = 0, what is the value of p?

Accepted Answer

5 A root makes the equation equal to zero. Substitute x = −4: 2(−4)² + p(−4) − 12 = 0, so 32 − 4p − 12 = 0. Thus 20 − 4p = 0, giving p = 5.

Introduction to Quadratic Equations

Practice questions

01 Which of the following quadratic equations has two equal and positive roots?

02 If \\(x+a)^2=x^2+10x+25\\) is an identity, what is the value of \\(a\\)?

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

04 If one root of (x^2+mx+16=0) is the reciprocal of the other, what can be said about (m)?

05 The roots of (x^2+px+9=0) are equal and negative. What is the value of (p)?

06 Which quadratic equation has sum of roots (0) and product (-49)?

07 If in \(2x^2+kx+18=0\), the product of roots is (-3) times the sum of roots, what is (k)?

08 What is the sum of cubes of the roots of (x^2-2x-3=0)?

09 If the roots of \\(x^2+bx+c=0\\) are \\(-2\\) and \\(7\\), what is the value of \\(b+c\\)?

10 Which of the following quadratic equations has one root \(x=0\) and the other root positive?

11 If \\(a=-1\\) is substituted in the equation \\((a+1)x^2+(a^2-1)x+2=0\\), what type of result is obtained?

12 What is the standard form of ((x-1)^2+(x-2)^2=(x-3)^2)?

13 If \(x=1\) is a root of the quadratic equation \(x^2+(2k-1)x+k=0\), what is the value of \(k\)?

14 The roots of the equation \(x^2-4x+1=0\) are \(\alpha\) and \(\beta\). What is the value of \(\alpha+\beta+\alpha\beta\)?

15 If the roots of \\(x^2+px+q=0\\) are 1 and \\(p\\), which relation for \\(q\\) is correct?

16 Which of the following is a quadratic equation in x but is not written in standard form?

17 Which equation will have a negative product of roots?

18 A right triangle has a base of \((x+1)\) units, a height of \((x+3)\) units, and an area of \(30\) square units. Which quadratic equation is obtained from this condition?

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

20 The equation \(x^2+4x+k=0\) has no real roots. What is the correct condition on \(k\)?

21 If the equation \(x^2-2ax+a^2-9=0\) is given, what is the difference between the larger and smaller roots?

22 What is the standard quadratic form of ((6x+1)(x-4)=5x)?

23 What is the standard form of the equation \\(2x-3\\)^2+(x+5)^2=34?

24 Write (\frac{x^2+1}{3}-\frac{x-2}{2}=5) in standard form with integer coefficients.

25 If x = -4 is a root of the equation 2x² + px − 12 = 0, what is the value of p?

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