Q: Which quadratic equation has roots whose sum is 5 and product is 6?

$x^2-5x+6=0$ For a monic quadratic with root-sum S and product P the equation is $x^2-Sx+P=0$. With S=5 and P=6 this gives $x^2-5x+6=0$, so option A is correct. Check distractors: B has product 6 but sum $-5$, while C and D have product 5, not 6. Exam tip: remember for $ax^2+bx+c=0$ the sum of roots = $-b/a)$ and product = $c/a)$.

Q: What is the product of the roots of the equation $3x^2+2x-8=0$?

$-\frac{8}{3}$ For a quadratic $ax^2+bx+c=0$, the product of roots equals $c/a$ (Vieta's formula). Here $a=3$ and $c=-8$, so the product is $\dfrac{-8}{3}=-\tfrac{8}{3}$. Option B ($\tfrac{8}{3}$) is a sign error; options C and D have incorrect numerical values. Exam tip: apply $c/a$ directly and pay attention to the sign of $c$ from the given equation.

Q: If the product of the roots of $kx^2-4x+2=0$ is 1, what is the value of $k$?

2 For a quadratic $ax^2+bx+c=0$, the product of the roots equals $\dfrac{c}{a}$. Here $a=k$ and $c=2$, so $\dfrac{2}{k}=1$ which gives $k=2$. Option 1 might arise from mistakenly taking $c$ alone instead of $c/a$; options -2 and 4 do not satisfy the relation $2/k=1$. Exam tip: always identify $a$ (coefficient of $x^2$) before applying the product formula $c/a$.

Q: In which equation is (x=1) not a root?

(x^2+x+1=0) Putting (x=1) gives (1+1+1=3\neq 0). To check when a value is not a root, use substitution too.

Q: If the roots of $x^2+px+16=0$ are 4 and 4, what is the value of $p$?

-8 The sum of the roots is 4+4 = 8. For a quadratic $ax^2+bx+c=0$, the sum of roots equals $-\frac{b}{a}$. Here $a=1, b=p$, so $-p=8$ and hence $p=-8$. The product check also matches: $4\times4=16=c/a$. The common wrong choice 8 comes from forgetting the negative sign in the relation. Exam tip: use sum = -b/a and product = c/a and verify both to avoid sign errors.

Question 1

If (x=2) and (x=-3) are roots of a quadratic equation, what is the sum of the roots?

Accepted Answer

(-1) The sum of roots is (2+(-3)=-1). When adding integers with unlike signs, take the sign of the larger magnitude.

Question 2

If the roots are (4) and (6), what is the product of the roots?

Accepted Answer

(24) The product of roots is (4\cdot6=24). Sum and product are both useful while forming an equation from roots.

Question 3

Which quadratic equation has roots whose sum is 5 and product is 6?

Accepted Answer

$x^2-5x+6=0$ For a monic quadratic with root-sum S and product P the equation is $x^2-Sx+P=0$. With S=5 and P=6 this gives $x^2-5x+6=0$, so option A is correct. Check distractors: B has product 6 but sum $-5$, while C and D have product 5, not 6. Exam tip: remember for $ax^2+bx+c=0$ the sum of roots = $-b/a)$ and product = $c/a)$.

Question 4

What is the sum of roots of (2x^2-5x+3=0)?

Accepted Answer

(\frac{5}{2}) The sum of roots is (-\frac{b}{a}=-\frac{-5}{2}=\frac{5}{2}). Pay attention to the sign of (b) in the formula.

Question 5

What is the product of the roots of the equation $3x^2+2x-8=0$?

Accepted Answer

$-\frac{8}{3}$ For a quadratic $ax^2+bx+c=0$, the product of roots equals $c/a$ (Vieta's formula). Here $a=3$ and $c=-8$, so the product is $\dfrac{-8}{3}=-\tfrac{8}{3}$. Option B ($\tfrac{8}{3}$) is a sign error; options C and D have incorrect numerical values. Exam tip: apply $c/a$ directly and pay attention to the sign of $c$ from the given equation.

Question 6

If the product of the roots of $kx^2-4x+2=0$ is 1, what is the value of $k$?

Accepted Answer

2 For a quadratic $ax^2+bx+c=0$, the product of the roots equals $\dfrac{c}{a}$. Here $a=k$ and $c=2$, so $\dfrac{2}{k}=1$ which gives $k=2$. Option 1 might arise from mistakenly taking $c$ alone instead of $c/a$; options -2 and 4 do not satisfy the relation $2/k=1$. Exam tip: always identify $a$ (coefficient of $x^2$) before applying the product formula $c/a$.

Question 7

In which equation is (x=1) not a root?

Accepted Answer

(x^2+x+1=0) Putting (x=1) gives (1+1+1=3
eq 0). To check when a value is not a root, use substitution too.

Question 8

If the roots of $x^2+px+16=0$ are 4 and 4, what is the value of $p$?

Accepted Answer

-8 The sum of the roots is 4+4 = 8. For a quadratic $ax^2+bx+c=0$, the sum of roots equals $-\frac{b}{a}$. Here $a=1, b=p$, so $-p=8$ and hence $p=-8$. The product check also matches: $4\times4=16=c/a$. The common wrong choice 8 comes from forgetting the negative sign in the relation. Exam tip: use sum = -b/a and product = c/a and verify both to avoid sign errors.

Question 9

Which quadratic equation is obtained by simplifying ((x+2)^2-(x+2)-6=0)?

Accepted Answer

(x^2+3x-4=0) To simplify the equation, first expand the square of the binomial. The identity $x+2)^2 = x^2 + 4x + 4$ is used because the middle term is twice the product of x and 2. The remaining terms are then combined like ordinary algebraic terms. Care is needed because the expression contains subtraction of both (x+2) and 6.

Substituting the expansion gives x^2 + 4x + 4 - x - 2 - 6 = 0. Combining like terms produces x^2 + 3x - 4 = 0. Therefore option A is correct. The coefficient of x is 3 because 4x - x = 3x, and the constant is -4 because 4 - 2 - 6 = -4. The other options result from expanding or combining the terms incorrectly.

Question 10

What is the standard form of ((x-1)(x+6)=2x)?

Accepted Answer

(x^2+3x-6=0) The left side gives (x^2+5x-6), and subtracting (2x) gives (x^2+3x-6=0). First expand and then bring all terms to one side.

Question 11

If the equation $x^2-9x+20=0$ is given and $x=4$ is substituted, what value does the left-hand side take?

Accepted Answer

0 Substitute directly: $4^2-9\cdot4+20=16-36+20=0$. Thus the left-hand side equals 0 and $x=4$ is a root of the equation. Option B (4) is incorrect because it confuses the variable's value with the value of the polynomial; the polynomial evaluates to 0, not 4. Exam tip: perform arithmetic step by step after substitution to avoid sign errors.

Question 12

Which equation will have roots that are opposite numbers?

Accepted Answer

(x^2-16=0) (x^2-16=0) gives (x=\pm4), which are opposite numbers. Pure quadratics often give opposite roots.

Question 13

In (x^2+kx+25=0), if the roots are equal and negative, which possible value of (k) is correct?

Accepted Answer

(10) For equal roots, (D=0) gives (k^2=100), and for equal negative roots (-\frac{k}{2}<0) is needed. Hence (k=10) is correct.

Question 14

In which option is the (x) term absent but the equation is still quadratic?

Accepted Answer

(x^2-49=0) In (x^2-49=0), the (x^2) term is present and the (x) term is absent. An equation can be quadratic even without an (x) term.

Question 15

If (3x^2-12x+12=0) is divided by (3), which simplified equation is obtained?

Accepted Answer

(x^2-4x+4=0) The direct answer is option A, $x^2-4x+4=0$. Dividing an equation by 3 means dividing every term by 3, not just one term: $3x^2/3=x^2$, $-12x/3=-4x$, and $12/3=4$. The right side remains $0/3=0$. Therefore the result is $x^2-4x+4=0$. Option A is correct, and the roots remain unchanged because 3 is non-zero. Option B, $x^2+4x+4=0$, has the wrong sign for the middle term; −12 divided by 3 is −4, not +4. Option C, $3x^2-4x+4=0$, fails to divide the first term by 3. Option D, $x^2-12x+12=0$, divides only the first term and leaves the other coefficients unchanged. A common mistake is to divide selected terms; always apply the operation to every term. Memory cue: divide each coefficient by the same non-zero number.

Question 16

Which pair of numbers helps determine the roots of the equation $x^2-11x+30=0$?

Accepted Answer

(5) and (6) We need two numbers whose product is 30 and whose sum is 11. $5\cdot6=30$ and $5+6=11$, so the quadratic factors as $(x-5)(x-6)$ and the roots are $x=5$ and $x=6$. The closest distractor, $3$ and $10$, also multiply to 30 but sum to 13, so it does not match the middle coefficient. Exam tip: For $ax^2+bx+c$, look for factor pairs of $c$ whose sum equals $b$, remembering to account for signs of the coefficients.

Question 17

If the roots of $x^2+ax+10=0$ are $-2$ and $-5$, what is the value of $a$?

Accepted Answer

7 Use Vieta's relations. For $x^2+ax+10=0$, the sum of roots equals $-a$ and the product equals $10$. The given roots sum to $-2)+(-5)=-7\, so \(-a=-7$ and hence $a=7$. Check the product: $(-2)(-5)=10$ matches the constant term, confirming consistency. The closest distractor $-7$ stems from reversing the sign convention (thinking sum equals $a$ instead of $-a$). Exam tip: For $x^2+bx+c=0$ remember sum = $-b$ and product = $c$ to avoid sign mistakes.

Question 18

The area of a square is 7 more than 6 times its side. If the side is $x$, which equation is correct?

Accepted Answer

$x^2-6x-7=0$ For a square with side $x$, area = $x^2$. The statement "7 more than 6 times its side" means area = $6x+7$, so $x^2=6x+7$. Rearranging gives $x^2-6x-7=0$, which matches option A. The closest distractor C ($x^2-7x-6=0$) would correspond to area = $7x+6$, which swaps the coefficients and is not what the question states. Option B has incorrect signs and option D has a different leading coefficient, so both are incorrect. Exam tip: Translate phrases like "a more than b times" as $b x + a$ to avoid sign or order mistakes.

Question 19

The equation $x^2-4x+4=0$ can be written in which perfect-square form?

Accepted Answer

(x-2)^2=0 Use the identity $(a-b)^2=a^2-2ab+b^2$. With $a=x, b=2$ we get $(x-2)^2=x^2-2\cdot x\cdot2+2^2=x^2-4x+4$, so the left-hand side is the perfect square $(x-2)^2$. The closest distractor, $(x+2)^2$, expands to $x^2+4x+4$ and therefore has the wrong sign on the middle term. The other options also do not match both the middle and constant terms. Exam tip: match the middle term as $\pm2ab$ to decide the sign in $(a\pm b)^2$.

Question 20

What is the standard form of ((x+3)^2=2x+15)?

Accepted Answer

(x^2+4x-6=0) Here ((x+3)^2=x^2+6x+9), and bringing all terms to one side gives (x^2+4x-6=0). Use the square identity and transposition carefully.

Question 21

What is the standard quadratic form of $(3x+2)(x-5)=0$?

Accepted Answer

$3x^2-13x-10=0$ Expanding $(3x+2)(x-5)$ gives $3x^2-15x+2x-10$. Combining like terms $-15x+2x=-13x$ yields the standard form $3x^2-13x-10=0$. Option C shows an intermediate grouping without combining the like x-terms; options B and D have incorrect signs or constants. Exam tip: always expand first, then combine like terms and write coefficients in descending powers of x.

Question 22

If ((2m-3)x^2+7x-1=0) is a quadratic equation, what is the correct condition for (m)?

Accepted Answer

(m
eq \frac{3}{2}) For a quadratic equation, the coefficient of (x^2) must not be (0). Thus (2m-3
eq 0), so (m
eq \frac{3}{2}).

Question 23

For the equation -4x^2 + 6x - 9 = 0, what is the value of a - b + c?

Accepted Answer

-19 Using the standard form ax^2 + bx + c = 0, the coefficients are a = -4, b = 6, c = -9. Compute a - b + c = (-4) - 6 + (-9) = -19. The choice -7 commonly results from mistakenly calculating a + b + c instead of a - b + c. Exam tip: write down a, b, c with their signs before substituting to avoid sign errors.

Question 24

What is the standard form of 8x − x² = 15 with a positive coefficient of x²?

Accepted Answer

x² − 8x + 15 = 0 A quadratic equation in standard form is ax² + bx + c = 0, and the question specifically requires the coefficient of x² to be positive. Start with 8x − x² = 15. Move 15 to the left: 8x − x² − 15 = 0, or equivalently −x² + 8x − 15 = 0. Since the x² coefficient is negative, multiply the entire equation by −1. This changes every sign and gives x² − 8x + 15 = 0. Therefore option A is correct. It is essential to change all terms, not just the x² term. Options B, C and D contain one or more incorrect signs and are not equivalent to the original equation.

Question 25

Is $x=-1$ a root of the equation $x^2+5x+4=0$?

Accepted Answer

Yes Substitute $x=-1$: $(-1)^2+5(-1)+4=1-5+4=0$. Therefore $x=-1$ is a root. Alternatively, factorization $x^2+5x+4=(x+1)(x+4)$ shows the roots are $x=-1$ and $x=-4$. Option C is incorrect because at $x=4$ the left side is $16+20+4\neq0$. Exam tip: verify roots by substitution or by factoring the quadratic.

Introduction to Quadratic Equations

Practice questions

01 If (x=2) and (x=-3) are roots of a quadratic equation, what is the sum of the roots?

02 If the roots are (4) and (6), what is the product of the roots?

03 Which quadratic equation has roots whose sum is 5 and product is 6?

04 What is the sum of roots of (2x^2-5x+3=0)?

05 What is the product of the roots of the equation \(3x^2+2x-8=0\)?

06 If the product of the roots of \(kx^2-4x+2=0\) is 1, what is the value of \(k\)?

07 In which equation is (x=1) not a root?

08 If the roots of \(x^2+px+16=0\) are 4 and 4, what is the value of \(p\)?

09 Which quadratic equation is obtained by simplifying ((x+2)^2-(x+2)-6=0)?

10 What is the standard form of ((x-1)(x+6)=2x)?

11 If the equation \(x^2-9x+20=0\) is given and \(x=4\) is substituted, what value does the left-hand side take?

12 Which equation will have roots that are opposite numbers?

13 In (x^2+kx+25=0), if the roots are equal and negative, which possible value of (k) is correct?

14 In which option is the (x) term absent but the equation is still quadratic?

15 If (3x^2-12x+12=0) is divided by (3), which simplified equation is obtained?

16 Which pair of numbers helps determine the roots of the equation \(x^2-11x+30=0\)?

17 If the roots of \(x^2+ax+10=0\) are \(-2\) and \(-5\), what is the value of \(a\)?

18 The area of a square is 7 more than 6 times its side. If the side is \(x\), which equation is correct?

19 The equation \(x^2-4x+4=0\) can be written in which perfect-square form?

20 What is the standard form of ((x+3)^2=2x+15)?

21 What is the standard quadratic form of \((3x+2)(x-5)=0\)?

22 If ((2m-3)x^2+7x-1=0) is a quadratic equation, what is the correct condition for (m)?

23 For the equation -4x^2 + 6x - 9 = 0, what is the value of a - b + c?

24 What is the standard form of 8x − x² = 15 with a positive coefficient of x²?

25 Is \(x=-1\) a root of the equation \(x^2+5x+4=0\)?

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