Q: If in $3x^2+kx+12=0$, the product of roots is (-2) times the sum of roots, what is (k)?

(6) The direct answer is A: $k=6$. For $3x^2+kx+12=0$, the product of roots is $12/3=4$, while their sum is $-k/3$. The condition says $4=-2$ times the sum, so $4=-2(-k/3)=2k/3$. Multiplying by 3 gives $12=2k$, hence $k=6$. Option A works exactly. Option B, $-6$, makes the sum 2 and -2 times it -4, not 4. Option C, 12, gives the right side 8. Option D, -12, gives -8. The signs come from the minus sign in the sum formula. Exam cue: write $S=-b/a$ and $P=c/a$ before substituting.

Q: If the roots of the quadratic equation $x^2+bx+c=0$ are $-3$ and $8$, what is the value of $b+c$?

-29 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 $-3+8=5$, so $-b=5$, giving $b=-5$. The product is $(-3)(8)=-24$, so $c=-24$. Therefore, $b+c=-5-24=-29$. Exam tip: For $x^2+bx+c=0$, use $b=-(\text{sum of roots})$ and $c=\text{product of roots}$.

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

$x^2+7x=0$ In option A, $x^2+7x=x(x+7)$. Hence, the roots are $x=0$ and $x=-7$, so the second root is negative. Option B has roots $0$ and $7$, and therefore does not satisfy the condition. Exam tip: For an equation of the form $x^2+bx=0$, the roots are $0$ and $-b$.

Q: If $a=2$, what type of statement does $(a-2)x^2+(a^2-4)x+5=0$ become?

Contradictory statement (no solution) Substituting $a=2$ gives $(2-2)x^2+(2^2-4)x+5=0$, or $0x^2+0x+5=0$, which reduces to $5=0$. This statement is false and is satisfied by no value of $x$, so it is a contradictory statement. It is not linear because the coefficient of $x$ is also zero, and it is not always true. Exam tip: After substituting a parameter value, check the highest non-zero power and whether the resulting constant statement is true or false.

Q: If $x=-2$ is a root of the equation $x^2+(3k+1)x+2k=0$, what is the value of $k$?

$\frac{1}{2}$ Since $x=-2$ is a root, substitute it into the equation: $(-2)^2+(3k+1)(-2)+2k=0$. This gives $4-6k-2+2k=0$, so $2-4k=0$ and hence $k=\frac{1}{2}$. The nearby distractor $k=-\frac{1}{2}$ does not make the equation equal to zero. Exam tip: when a root is given, substitute it directly into the polynomial equation.

Q: The roots of the equation \$x^2-5x+2=0\$ are \$\\alpha\$ and \$\\beta\$. What is the value of \$\\alpha+\\beta+2\\alpha\\beta\$?

9 For a quadratic equation \$ax^2+bx+c=0\$, the sum of the roots is \$ -b/a \$ and their product is \$c/a\$. Here, \$a=1,b=-5,c=2\$, so \$\\alpha+\\beta=5\$ and \$\\alpha\\beta=2\$. Therefore, \$\\alpha+\\beta+2\\alpha\\beta=5+2(2)=9\$. Exam tip: use the sum and product of roots directly instead of solving for the individual roots.

Question 1

If in $3x^2+kx+12=0$, the product of roots is (-2) times the sum of roots, what is (k)?

Accepted Answer

(6) The direct answer is A: $k=6$. For $3x^2+kx+12=0$, the product of roots is $12/3=4$, while their sum is $-k/3$. The condition says $4=-2$ times the sum, so $4=-2(-k/3)=2k/3$. Multiplying by 3 gives $12=2k$, hence $k=6$. Option A works exactly. Option B, $-6$, makes the sum 2 and -2 times it -4, not 4. Option C, 12, gives the right side 8. Option D, -12, gives -8. The signs come from the minus sign in the sum formula. Exam cue: write $S=-b/a$ and $P=c/a$ before substituting.

Question 2

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

Accepted Answer

(117) The sum of roots is (3) and the product is (-10). (\alpha^3+\beta^3=(\alpha+\beta)^3-3\alpha\beta(\alpha+\beta)=27+90=117).

Question 3

If the roots of the quadratic equation $x^2+bx+c=0$ are $-3$ and $8$, what is the value of $b+c$?

Accepted Answer

-29 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 $-3+8=5$, so $-b=5$, giving $b=-5$. The product is $(-3)(8)=-24$, so $c=-24$. Therefore, $b+c=-5-24=-29$. Exam tip: For $x^2+bx+c=0$, use $b=-(\text{sum of roots})$ and $c=\text{product of roots}$.

Question 4

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

Accepted Answer

$x^2+7x=0$ In option A, $x^2+7x=x(x+7)$. Hence, the roots are $x=0$ and $x=-7$, so the second root is negative. Option B has roots $0$ and $7$, and therefore does not satisfy the condition. Exam tip: For an equation of the form $x^2+bx=0$, the roots are $0$ and $-b$.

Question 5

If $a=2$, what type of statement does $(a-2)x^2+(a^2-4)x+5=0$ become?

Accepted Answer

Contradictory statement (no solution) Substituting $a=2$ gives $(2-2)x^2+(2^2-4)x+5=0$, or $0x^2+0x+5=0$, which reduces to $5=0$. This statement is false and is satisfied by no value of $x$, so it is a contradictory statement. It is not linear because the coefficient of $x$ is also zero, and it is not always true. Exam tip: After substituting a parameter value, check the highest non-zero power and whether the resulting constant statement is true or false.

Question 6

What is the standard form of ((x+2)^2+(x-5)^2=(x+1)^2)?

Accepted Answer

(x^2-8x+28=0) Expanding gives left side (2x^2-6x+29) and right side (x^2+2x+1). Subtracting gives (x^2-8x+28=0).

Question 7

If $x=-2$ is a root of the equation $x^2+(3k+1)x+2k=0$, what is the value of $k$?

Accepted Answer

$\frac{1}{2}$ Since $x=-2$ is a root, substitute it into the equation: $(-2)^2+(3k+1)(-2)+2k=0$. This gives $4-6k-2+2k=0$, so $2-4k=0$ and hence $k=\frac{1}{2}$. The nearby distractor $k=-\frac{1}{2}$ does not make the equation equal to zero. Exam tip: when a root is given, substitute it directly into the polynomial equation.

Question 8

The roots of the equation $x^2-5x+2=0$ are $\alpha$ and $\beta$. What is the value of $\alpha+\beta+2\alpha\beta$?

Accepted Answer

9 For a quadratic equation $ax^2+bx+c=0$, the sum of the roots is $ -b/a $ and their product is $c/a$. Here, $a=1,b=-5,c=2$, so $\alpha+\beta=5$ and $\alpha\beta=2$. Therefore, $\alpha+\beta+2\alpha\beta=5+2(2)=9$. Exam tip: use the sum and product of roots directly instead of solving for the individual roots.

Question 9

If the roots of the equation x² + px + q = 0 are 2 and p, which relation involving q is correct?

Accepted Answer

q = 2p For a monic quadratic, the product of the roots equals its constant term. Thus q = 2 × p = 2p. The sum of the roots also gives 2 + p = −p, so p = −1 and q = −2; this is consistent with q = 2p.

Question 10

If α and β are the roots of x² − 9x + 20 = 0, what is the value of (α − 4)(β − 4)?

Accepted Answer

0 For this equation, the sum of the roots is α + β = 9 and their product is αβ = 20. Therefore, (α − 4)(β − 4) = αβ − 4(α + β) + 16 = 20 − 36 + 16 = 0.

Question 11

A right triangle has a base of (x + 2) units and a height of (x + 6) units. If its area is 40 square units, which quadratic equation in x is correct?

Accepted Answer

x² + 8x − 68 = 0 The area of a triangle is half the product of its base and height: ½(x + 2)(x + 6) = 40. Multiplying by 2 gives (x + 2)(x + 6) = 80. Expanding and rearranging gives x² + 8x + 12 − 80 = 0, or x² + 8x − 68 = 0.

Question 12

If α and β are the roots of x² − 9x + 14 = 0, what is the value of (α + β)²?

Accepted Answer

81 For ax² + bx + c = 0, the sum of the roots is −b/a. Here, a = 1 and b = −9, so α + β = 9. Therefore, (α + β)² = 9² = 81.

Question 13

If (\alpha,\beta) are roots of (x^2-4x-21=0), what is (\alpha\beta+2\alpha+2\beta)?

Accepted Answer

(-13) The direct answer is option A: -13. For a monic quadratic x² + bx + c = 0 with roots α and β, the sum of roots is α + β = -b and the product is αβ = c. In x² - 4x - 21 = 0, b = -4 and c = -21. Therefore α + β = -(-4) = 4 and αβ = -21. The required expression is αβ + 2α + 2β. Group the last two terms: αβ + 2(α + β). Substitute the known values: -21 + 2(4) = -21 + 8 = -13. Thus option A is correct. Option B, -21, is only the product αβ and ignores the additional terms. Option C, 13, has the wrong sign and does not follow from the calculation. Option D, 29, is obtained by an incorrect operation, such as adding quantities with the wrong sign. The useful lesson is to identify the sum and product first rather than solving the quadratic roots individually. Memory cue: for x² + bx + c, remember “sum opposite of b, product c.”

Question 14

The equation $x^2+8x+k=0$ has no real roots. What is the correct condition on $k$?

Accepted Answer

$k>16$ A quadratic equation has no real roots only when its discriminant $D=b^2-4ac$ is negative. Here, $a=1, b=8, c=k$, so $D=8^2-4(1)(k)=64-4k$. Thus, $64-4k<0$, which gives $k>16$. Note that $k=16$ makes the discriminant zero and produces one repeated real root, so it is not correct.

Question 15

If the equation $x^2-2ax+a^2-16=0$ is given, what is the difference between its roots?

Accepted Answer

8 The equation can be rewritten as $(x-a)^2-16=0$. Thus, $(x-a)^2=16$, giving the roots $x=a+4$ and $x=a-4$. Their difference is $(a+4)-(a-4)=8$. The value 4 represents the distance of each root from $a$, not the distance between the two roots. In the exam, completing the square is a quick method for such questions.

Question 16

If the roots of (x^2+px+q=0) are (3) and (p), what is a possible value of (p)?

Accepted Answer

(-\frac{3}{2}) The sum of roots is (3+p), and in the equation the sum is (-p). Thus (3+p=-p), giving (p=-\frac{3}{2}).

Question 17

What is the standard form of $(x-2)(x+3)+(2x-1)(x-4)=0$?

Accepted Answer

$3x^2-8x-2=0$ Expanding the two products gives $(x-2)(x+3)=x^2+x-6$ and $(2x-1)(x-4)=2x^2-9x+4$. Combining like terms, $x^2+2x^2=3x^2$, $x-9x=-8x$, and $-6+4=-2$. Hence the standard form is $3x^2-8x-2=0$. Option B has the wrong sign for the linear term. Exam tip: expand each bracket carefully, then combine the quadratic, linear, and constant terms separately.

Question 18

What is the standard quadratic form of (x + 2)/(x − 1) + (x − 1)/(x + 2) = 5/2?

Accepted Answer

x² + x − 20 = 0 The original equation requires x ≠ 1 and x ≠ −2 because these values make a denominator zero. Multiply both sides by 2(x − 1)(x + 2), which is valid under those restrictions. This gives 2(x + 2)² + 2(x − 1)² = 5(x − 1)(x + 2). Expanding the left side: 2(x² + 4x + 4) + 2(x² − 2x + 1) = 4x² + 4x + 10. Expanding the right side gives 5(x² + x − 2) = 5x² + 5x − 10. Moving the left side to the right yields 0 = x² + x − 20, or x² + x − 20 = 0. Thus option A is correct; the other options have an incorrect sign, constant, or leading coefficient.

Introduction to Quadratic Equations

Practice questions

01 If in \(3x^2+kx+12=0\), the product of roots is (-2) times the sum of roots, what is (k)?

02 What is the sum of cubes of the roots of (x^2-3x-10=0)?

03 If the roots of the quadratic equation \(x^2+bx+c=0\) are \(-3\) and \(8\), what is the value of \(b+c\)?

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

05 If \(a=2\), what type of statement does \((a-2)x^2+(a^2-4)x+5=0\) become?

06 What is the standard form of ((x+2)^2+(x-5)^2=(x+1)^2)?

07 If \(x=-2\) is a root of the equation \(x^2+(3k+1)x+2k=0\), what is the value of \(k\)?

08 The roots of the equation \\(x^2-5x+2=0\\) are \\(\\alpha\\) and \\(\\beta\\). What is the value of \\(\\alpha+\\beta+2\\alpha\\beta\\)?

09 If the roots of the equation x² + px + q = 0 are 2 and p, which relation involving q is correct?

10 If α and β are the roots of x² − 9x + 20 = 0, what is the value of (α − 4)(β − 4)?

11 A right triangle has a base of (x + 2) units and a height of (x + 6) units. If its area is 40 square units, which quadratic equation in x is correct?

12 If α and β are the roots of x² − 9x + 14 = 0, what is the value of (α + β)²?

13 If (\alpha,\beta) are roots of (x^2-4x-21=0), what is (\alpha\beta+2\alpha+2\beta)?

14 The equation \(x^2+8x+k=0\) has no real roots. What is the correct condition on \(k\)?

15 If the equation \(x^2-2ax+a^2-16=0\) is given, what is the difference between its roots?

16 If the roots of (x^2+px+q=0) are (3) and (p), what is a possible value of (p)?

17 What is the standard form of \((x-2)(x+3)+(2x-1)(x-4)=0\)?

18 What is the standard quadratic form of (x + 2)/(x − 1) + (x − 1)/(x + 2) = 5/2?

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