Q: If (x=-2) is not a root of (kx^2+5x+6=0), what condition must (k) satisfy?

(k\neq1) Putting (x=-2), the left side becomes (4k-10+6=4k-4). For it not to be a root, (4k-4\neq0), so (k\neq1).

Q: Which quadratic equation has sum of roots (0) and product (-144)?

(x^2-144=0) The monic equation is (x^2-(\text{sum})x+\text{product}=0). Using sum (0) and product (-144) gives (x^2-144=0).

Q: If in $5x^2+kx+45=0$, the product of roots is (-3) times the sum of roots, what is (k)?

(15) The product is (9) and the sum is $-\frac{k}{5}$. From $9=-3\left(-\frac{k}{5}\right)$, we get $k=15$.

Q: If the roots of the quadratic equation $x^2+bx+c=0$ are $-5$ and $11$, what is the value of $b+c$?

-61 For the quadratic equation $x^2+bx+c=0$, the sum of the roots is $-b$ and their product is $c$. Here, the sum is $-5+11=6$, so $-b=6$, giving $b=-6$. Their product is $(-5)(11)=-55$, so $c=-55$. Therefore, $b+c=-6-55=-61$. Exam tip: In a monic quadratic, use the root relations sum $=-b$ and product $=c$ directly.

Q: If $x=-2$ is a root of the equation $x^2+(5k-2)x+4k=0$, what is the value of $k$?

$\frac{4}{3}$ Since $x=-2$ is a root, substituting it into the equation gives $(-2)^2+(5k-2)(-2)+4k=0$. Therefore, $4-10k+4+4k=0$, so $8-6k=0$ and $k=\frac{4}{3}$. Hence, option D is correct. Exam tip: When a root is given, substitute it directly into the quadratic equation and solve for the parameter.

Q: The roots of the equation $x^2-8x+5=0$ are $\alpha$ and $\beta$. What is the value of $\alpha+\beta+4\alpha\beta$?

28 By Vieta’s formulas for $x^2-8x+5=0$, the sum of the roots is $\alpha+\beta=8$ and their product is $\alpha\beta=5$. Therefore, $\alpha+\beta+4\alpha\beta=8+4(5)=28$. Option 13 results from adding the product only once, so it is incorrect. Exam tip: for $ax^2+bx+c=0$, the sum of roots is $-b/a$ and the product is $c/a$.

Question 1

The roots of (x^2+px+36=0) are equal and positive. What is the value of (p)?

Accepted Answer

(-12) For equal roots, (p^2-144=0) gives (p=\pm12). For equal positive roots, (-\frac{p}{2}>0), so (p=-12).

Question 2

If (x=-2) is not a root of (kx^2+5x+6=0), what condition must (k) satisfy?

Accepted Answer

(k
eq1) Putting (x=-2), the left side becomes (4k-10+6=4k-4). For it not to be a root, (4k-4
eq0), so (k
eq1).

Question 3

Which quadratic equation has sum of roots (0) and product (-144)?

Accepted Answer

(x^2-144=0) The monic equation is (x^2-(	ext{sum})x+	ext{product}=0). Using sum (0) and product (-144) gives (x^2-144=0).

Question 4

If in $5x^2+kx+45=0$, the product of roots is (-3) times the sum of roots, what is (k)?

Accepted Answer

(15) The product is (9) and the sum is $-\frac{k}{5}$. From $9=-3\left(-\frac{k}{5}\right)$, we get $k=15$.

Question 5

What is the sum of cubes of the roots of (x^2-6x-16=0)?

Accepted Answer

(504) The sum of roots is (6) and the product is (-16). (\alpha^3+\beta^3=(\alpha+\beta)^3-3\alpha\beta(\alpha+\beta)=216+288=504).

Question 6

If the roots of the quadratic equation $x^2+bx+c=0$ are $-5$ and $11$, what is the value of $b+c$?

Accepted Answer

-61 For the quadratic equation $x^2+bx+c=0$, the sum of the roots is $-b$ and their product is $c$. Here, the sum is $-5+11=6$, so $-b=6$, giving $b=-6$. Their product is $(-5)(11)=-55$, so $c=-55$. Therefore, $b+c=-6-55=-61$. Exam tip: In a monic quadratic, use the root relations sum $=-b$ and product $=c$ directly.

Question 7

If a = 3 is substituted, what type of statement does (a − 3)x² + (a² − 9)x + 11 = 0 become?

Accepted Answer

Contradictory statement Substituting a = 3 gives a − 3 = 0 and a² − 9 = 9 − 9 = 0. Therefore, the equation reduces to 11 = 0, which is a false and contradictory statement, so it has no solution. Exam tip: When the coefficients of the variable terms become zero, check whether the remaining constant statement is true or false; a false statement has no solution.

Question 8

What is the standard form of ((x+5)^2+(x-6)^2=(x+2)^2)?

Accepted Answer

(x^2-10x+57=0) Expanding gives left side (2x^2-2x+61) and right side (x^2+4x+4). Subtracting gives (x^2-6x+57=0).

Question 9

If $x=-2$ is a root of the equation $x^2+(5k-2)x+4k=0$, what is the value of $k$?

Accepted Answer

$\frac{4}{3}$ Since $x=-2$ is a root, substituting it into the equation gives $(-2)^2+(5k-2)(-2)+4k=0$. Therefore, $4-10k+4+4k=0$, so $8-6k=0$ and $k=\frac{4}{3}$. Hence, option D is correct. Exam tip: When a root is given, substitute it directly into the quadratic equation and solve for the parameter.

Question 10

The roots of the equation $x^2-8x+5=0$ are $\alpha$ and $\beta$. What is the value of $\alpha+\beta+4\alpha\beta$?

Accepted Answer

28 By Vieta’s formulas for $x^2-8x+5=0$, the sum of the roots is $\alpha+\beta=8$ and their product is $\alpha\beta=5$. Therefore, $\alpha+\beta+4\alpha\beta=8+4(5)=28$. Option 13 results from adding the product only once, so it is incorrect. Exam tip: for $ax^2+bx+c=0$, the sum of roots is $-b/a$ and the product is $c/a$.

Question 11

If the roots of (x^2+px+q=0) are (5) and (p), what is the value of (p)?

Accepted Answer

-(\frac{5}{2}) / (-\frac{5}{2}) The sum of roots is (5+p), and in the equation the sum is (-p). Thus (5+p=-p), giving (p=-\frac{5}{2}).

Question 12

If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2-16x+63=0$, what is the value of $(\alpha-7)(\beta-7)$?

Accepted Answer

$0$ By Vieta’s formulas, $\alpha+\beta=16$ and $\alpha\beta=63$. Therefore, $(\alpha-7)(\beta-7)=\alpha\beta-7(\alpha+\beta)+49=63-7(16)+49=0$. Hence, option A is correct. Exam tip: Use the sum and product of the roots directly instead of finding the roots separately.

Question 13

A right triangle has a base of (x+4) units, a height of (x+8) units, and an area of 72 square units. Which quadratic equation is formed from this information?

Accepted Answer

$x^2+12x-112=0$ The area of a triangle is $\frac{1}{2}\times\text{base}\times\text{height}$. Hence, $\frac{1}{2}(x+4)(x+8)=72$, so $(x+4)(x+8)=144$. Expanding gives $x^2+12x+32=144$, and therefore $x^2+12x-112=0$. Option B incorrectly subtracts $72$ instead of $144$, failing to account for the factor $\frac{1}{2}$. Exam tip: In triangle-area problems, do not forget the factor $\frac{1}{2}$.

Question 14

If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2-15x+44=0$, what is the value of $(\alpha+\beta)^2$?

Accepted Answer

225 By Vieta’s formula, the sum of the roots of $ax^2+bx+c=0$ is $-\frac{b}{a}$. Here, $a=1$ and $b=-15$, so $\alpha+\beta=-\frac{-15}{1}=15$. Therefore, $(\alpha+\beta)^2=15^2=225$. The value 44 is the product $\alpha\beta$, not the sum of the roots. Exam tip: Identify the sum and product of roots directly using Vieta’s formulas.

Question 15

After simplification, which of the following equations does not remain a quadratic equation in x?

Accepted Answer

$(x+1)^2=x^2+4x$ In option C, expanding gives $x^2+2x+1=x^2+4x$. The $x^2$ terms cancel, leaving $2x-1=0$, which is linear. Exam tip: simplify fully and then check the highest power of x.

Question 16

If the equation $x^2-2ax+a^2-36=0$ is given, what is the difference between its larger and smaller roots?

Accepted Answer

12 The equation can be rewritten as $(x-a)^2-36=0$. Thus, $(x-a)^2=36$, giving the roots $x=a+6$ and $x=a-6$. Their difference is $(a+6)-(a-6)=12$. The expression $a+6$ represents only one root, not the difference. Exam tip: First rewrite such equations in perfect-square form.

Question 17

If (\alpha,\beta) are roots of (2x^2-7x+3=0), what is the value of (\frac{\alpha}{\beta}+\frac{\beta}{\alpha})?

Accepted Answer

( \frac{37}{6} ) We use (\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}). Here (\alpha+\beta=\frac{7}{2}) and (\alpha\beta=\frac{3}{2}), so the value is (\frac{37}{6}).

Question 18

If the roots of (x^2-(2k+3)x+(k^2+3k)=0) are consecutive positive integers, what is the value of (k)?

Accepted Answer

(3) The sum is (2k+3) and the product is (k^2+3k). Direct checking with the options shows (k=3) gives the required equation pattern.

Question 19

If roots of (x^2+px+q=0) are (p+1) and (q-1), and (p+q=5), what is the value of (pq)?

Accepted Answer

(6) The sum of roots is (-p), so ((p+1)+(q-1)=p+q=-p). Using (p+q=5), the option consistent with the conditions is (6).

Question 20

What is the standard quadratic form of ((4x-3)(x+5)=2(3x-1))?

Accepted Answer

(4x^2+11x-13=0) Here ((4x-3)(x+5)=4x^2+17x-15) and (2(3x-1)=6x-2). Bringing all terms to one side gives (4x^2+11x-13=0).

Question 21

If ((t^2-64)x^2+(t-8)x+5=0) is a quadratic equation, what is the correct condition on (t)?

Accepted Answer

(t
eq \pm8) For the equation to be quadratic, the coefficient of (x^2) must not be (0). Here (t^2-64
eq0), so (t
eq\pm8).

Question 22

What is the standard form of the equation $3(x+1)^2+2(x-4)^2=74$?

Accepted Answer

5x^2-10x-39=0 Expand the squares first: $3(x+1)^2=3(x^2+2x+1)=3x^2+6x+3$ and $2(x-4)^2=2(x^2-8x+16)=2x^2-16x+32$. Adding gives $5x^2-10x+35=74$. Bring 74 to the left and simplify: $5x^2-10x+35-74=0$, so $5x^2-10x-39=0$. Option B is the closest distractor because it has the wrong sign on the linear term (+10x instead of -10x). Exam tip: expand carefully, combine like terms, then move the constant from the RHS to the LHS and re-check the arithmetic for the constant term (here $3+32-74=-39$).

Question 23

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

Accepted Answer

(16x^2+5x-74=0) Multiplying the whole equation by (20) gives (16x^2-12+5x-10=60). Therefore the standard form is (16x^2+5x-82=0).

Question 24

If $x=-6$ is a root of $2x^2+px-18=0$, what is the value of $p$?

Accepted Answer

9 Substitute $x=-6$ into $2x^2+px-18=0$: $2(-6)^2 + p(-6) -18 = 0$. This gives $72 - 6p - 18 = 0$ → $54 - 6p = 0$, so $p = 54/6 = 9$. The choice $-9$ typically arises from a sign error (treating $p(-6)$ as $+6p$). Exam tip: always substitute the root and simplify step-by-step, checking signs carefully.

Question 25

Which monic quadratic equation has sum of roots (-15) and product (56)?

Accepted Answer

(x^2+15x+56=0) A monic equation is (x^2-(	ext{sum})x+	ext{product}=0). Substituting sum (-15) gives (x^2+15x+56=0).

Introduction to Quadratic Equations

Practice questions

01 The roots of (x^2+px+36=0) are equal and positive. What is the value of (p)?

02 If (x=-2) is not a root of (kx^2+5x+6=0), what condition must (k) satisfy?

03 Which quadratic equation has sum of roots (0) and product (-144)?

04 If in \(5x^2+kx+45=0\), the product of roots is (-3) times the sum of roots, what is (k)?

05 What is the sum of cubes of the roots of (x^2-6x-16=0)?

06 If the roots of the quadratic equation \(x^2+bx+c=0\) are \(-5\) and \(11\), what is the value of \(b+c\)?

07 If a = 3 is substituted, what type of statement does (a − 3)x² + (a² − 9)x + 11 = 0 become?

08 What is the standard form of ((x+5)^2+(x-6)^2=(x+2)^2)?

09 If \(x=-2\) is a root of the equation \(x^2+(5k-2)x+4k=0\), what is the value of \(k\)?

10 The roots of the equation \(x^2-8x+5=0\) are \(\alpha\) and \(\beta\). What is the value of \(\alpha+\beta+4\alpha\beta\)?

11 If the roots of (x^2+px+q=0) are (5) and (p), what is the value of (p)?

12 If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(x^2-16x+63=0\), what is the value of \((\alpha-7)(\beta-7)\)?

13 A right triangle has a base of (x+4) units, a height of (x+8) units, and an area of 72 square units. Which quadratic equation is formed from this information?

14 If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(x^2-15x+44=0\), what is the value of \((\alpha+\beta)^2\)?

15 After simplification, which of the following equations does not remain a quadratic equation in x?

16 If the equation \(x^2-2ax+a^2-36=0\) is given, what is the difference between its larger and smaller roots?

17 If (\alpha,\beta) are roots of (2x^2-7x+3=0), what is the value of (\frac{\alpha}{\beta}+\frac{\beta}{\alpha})?

18 If the roots of (x^2-(2k+3)x+(k^2+3k)=0) are consecutive positive integers, what is the value of (k)?

19 If roots of (x^2+px+q=0) are (p+1) and (q-1), and (p+q=5), what is the value of (pq)?

20 What is the standard quadratic form of ((4x-3)(x+5)=2(3x-1))?

21 If ((t^2-64)x^2+(t-8)x+5=0) is a quadratic equation, what is the correct condition on (t)?

22 What is the standard form of the equation \(3(x+1)^2+2(x-4)^2=74\)?

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

24 If \(x=-6\) is a root of \(2x^2+px-18=0\), what is the value of \(p\)?

25 Which monic quadratic equation has sum of roots (-15) and product (56)?

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