Q: If the product of roots of (px^2+6x+9=0) is (3), what is the value of (p)?

(3) Direct answer: Option A, p=3. For a quadratic equation \$ax^2+bx+c=0\$, the product of roots is \$c/a\$. Here a=p and c=9, so the product is \$9/p\$. The question says this product equals 3: \$9/p=3\$. Multiplying by p gives \$9=3p\$, and dividing by 3 gives \$p=3\$. Option A is correct. Option B, 6, would give product \$9/6\$, not 3. Option C, 9, would give product 1. Option D, -3, would give product -3, not the stated positive value. Also p cannot be zero because then the equation would not be quadratic and division by p would be invalid. Remember: product of roots is constant term divided by the coefficient of the squared term.

Q: Which pair is the correct roots of $x^2+2x-35=0$?

(5, -7) Factorise: $x^2+2x-35=(x+7)(x-5)=0$. Hence the roots are $x=-7$ and $x=5$, i.e. the pair $(5,-7)$. Check: sum $5+(-7)=-2=-b/a$ and product $5\times(-7)=-35=c/a$. Option B $(7,-5)$ has the correct product but wrong sum (+2), so it is incorrect. Exam tip: verify roots quickly using sum and product of roots or factorisation before finalising the answer.

Q: Which quadratic equation is obtained by simplifying $(x-3)^2+2(x-3)-8=0$?

$x^2-4x-5=0$ Expand and combine like terms: $(x-3)^2=x^2-6x+9$ and $2(x-3)=2x-6$. Adding and subtracting 8 gives $x^2-6x+9+2x-6-8=x^2-4x-5$. Therefore $x^2-4x-5=0$ is correct. Closest distractor D has the same linear term but wrong constant sign (+5 instead of -5). Exam tip: Always expand $(x-a)^2$ as $x^2-2ax+a^2$ and then combine like terms carefully.

Q: What is the standard form of $(x+2)(x-7)=3x$?

$x^2-8x-14=0$ First expand the left side: $(x+2)(x-7)=x^2-7x+2x-14=x^2-5x-14$. To put in standard form $ax^2+bx+c=0$, subtract $3x$ from both sides: $x^2-5x-14-3x=0\Rightarrow x^2-8x-14=0$. Option B ($x^2-5x-14=0$) is the expansion before moving $3x$, so it is the closest distractor. Options C and D result from sign or arithmetic mistakes. Exam tip: Expand first, then bring all terms to one side and order by descending powers of $x$.

Q: If $x^2-12x+35=0$, what is the value of the left-hand side when $x=5$?

0 Substitute $x=5$: the left-hand side becomes $5^2-12\cdot5+35=25-60+35=0$. Thus the value is 0 and $x=5$ is a root. Closest distractors are incorrect because they are either the substituted value (5) or give nonzero results when substituted; always verify by direct substitution and careful arithmetic. Exam tip: perform the arithmetic step-by-step to avoid sign mistakes.

Question 1

If the product of roots of (px^2+6x+9=0) is (3), what is the value of (p)?

Accepted Answer

(3) Direct answer: Option A, p=3. For a quadratic equation $ax^2+bx+c=0$, the product of roots is $c/a$. Here a=p and c=9, so the product is $9/p$. The question says this product equals 3: $9/p=3$. Multiplying by p gives $9=3p$, and dividing by 3 gives $p=3$. Option A is correct. Option B, 6, would give product $9/6$, not 3. Option C, 9, would give product 1. Option D, -3, would give product -3, not the stated positive value. Also p cannot be zero because then the equation would not be quadratic and division by p would be invalid. Remember: product of roots is constant term divided by the coefficient of the squared term.

Question 2

In which of the following equations is (x = -2) not a root?

Accepted Answer

x^2-2x+4=0 Check each equation by substituting x = -2.
A: (-2)^2 + 2(-2) = 4 - 4 = 0 → x = -2 is a root.
B: (-2)^2 + 5(-2) + 6 = 4 - 10 + 6 = 0 → x = -2 is a root.
C: 2(-2)^2 + 3(-2) - 2 = 8 - 6 - 2 = 0 → x = -2 is a root (this is the trickiest distractor because coefficients look nontrivial).
D: (-2)^2 - 2(-2) + 4 = 4 + 4 + 4 = 12 ≠ 0 → x = -2 is not a root.
Therefore option D is correct. Exam tip: For testing a single candidate root, direct substitution is quicker and less error-prone than factoring.

Question 3

Which pair is the correct roots of $x^2+2x-35=0$?

Accepted Answer

(5, -7) Factorise: $x^2+2x-35=(x+7)(x-5)=0$. Hence the roots are $x=-7$ and $x=5$, i.e. the pair $(5,-7)$. Check: sum $5+(-7)=-2=-b/a$ and product $5\times(-7)=-35=c/a$. Option B $(7,-5)$ has the correct product but wrong sum (+2), so it is incorrect. Exam tip: verify roots quickly using sum and product of roots or factorisation before finalising the answer.

Question 4

If the roots of $x^2+rx+36=0$ are 6 and 6, what is the value of $r$?

Accepted Answer

-12 If the roots are 6 and 6, their sum is 6+6=12 and product is 6×6=36. For the quadratic $x^2+rx+36=0$, the sum of roots equals $-r$ and the product equals 36. Hence $-r=12$ gives $r=-12$. Option A (12) mistakes the sign. Options C and D are the root values themselves, not the coefficient asked for. Exam tip: Always apply sum = -b/a and product = c/a, and check signs carefully.

Question 5

Which quadratic equation is obtained by simplifying $(x-3)^2+2(x-3)-8=0$?

Accepted Answer

$x^2-4x-5=0$ Expand and combine like terms: $(x-3)^2=x^2-6x+9$ and $2(x-3)=2x-6$. Adding and subtracting 8 gives $x^2-6x+9+2x-6-8=x^2-4x-5$. Therefore $x^2-4x-5=0$ is correct. Closest distractor D has the same linear term but wrong constant sign (+5 instead of -5). Exam tip: Always expand $(x-a)^2$ as $x^2-2ax+a^2$ and then combine like terms carefully.

Question 6

What is the standard form of $(x+2)(x-7)=3x$?

Accepted Answer

$x^2-8x-14=0$ First expand the left side: $(x+2)(x-7)=x^2-7x+2x-14=x^2-5x-14$. To put in standard form $ax^2+bx+c=0$, subtract $3x$ from both sides: $x^2-5x-14-3x=0\Rightarrow x^2-8x-14=0$. Option B ($x^2-5x-14=0$) is the expansion before moving $3x$, so it is the closest distractor. Options C and D result from sign or arithmetic mistakes. Exam tip: Expand first, then bring all terms to one side and order by descending powers of $x$.

Question 7

If $x^2-12x+35=0$, what is the value of the left-hand side when $x=5$?

Accepted Answer

0 Substitute $x=5$: the left-hand side becomes $5^2-12\cdot5+35=25-60+35=0$. Thus the value is 0 and $x=5$ is a root. Closest distractors are incorrect because they are either the substituted value (5) or give nonzero results when substituted; always verify by direct substitution and careful arithmetic. Exam tip: perform the arithmetic step-by-step to avoid sign mistakes.

Question 8

If the roots of (x^2-14x+k=0) are equal, what is (k)?

Accepted Answer

(49) Direct answer: Option B, k=49. Equal roots occur when the discriminant is zero. For $ax^2+bx+c=0$, the discriminant is $D=b^2-4ac$. Here a=1, b=-14, and c=k, so $D=(-14)^2-4(1)(k)=196-4k$. Set it equal to zero: $196-4k=0$, hence $4k=196$ and $k=49$. Option B is correct. Option A, 14, does not make the discriminant zero. Option C, 98, gives a negative discriminant, so the roots are not equal real roots. Option D, 196, also fails the zero-discriminant condition. The value 49 also agrees with the repeated-root form $x^2-14x+49=(x-7)^2$.

Question 9

In (x^2+kx+49=0), if the roots are equal and positive, which possible value of (k) is correct?

Accepted Answer

(-14) For equal roots, (k^2=196), and the equal root is (-\frac{k}{2}). For a positive root, (k=-14) is correct.

Question 10

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

Accepted Answer

(3x^2-27=0) In (3x^2-27=0), the (x^2) term is present and the (x) term is absent. An equation can be quadratic even without the (x) term.

Question 11

If (6x^2-18x+12=0) is divided by (6), which simplified equation is obtained?

Accepted Answer

(x^2-3x+2=0) Dividing every term by (6) gives (x^2-3x+2=0). Dividing by a common nonzero factor does not change the roots.

Question 12

Which pair of integers is useful for factoring the quadratic equation x^2 - 13x + 42 = 0?

Accepted Answer

6 and 7 We need two integers m and n such that m·n = 42 and m + n = 13. Checking 6 and 7 gives 6·7 = 42 and 6+7 = 13, so the quadratic factors as (x-6)(x-7), which expands to x^2 - 13x + 42. A close distractor is 3 and 14: their product is 42 but 3+14 = 17, not 13, so they do not produce the correct middle term. Exam tip: list factor pairs of the constant term and check which pair sums to the coefficient of x (taking sign into account).

Question 13

If the quadratic equation $x^2 + s x + 18 = 0$ has roots $-3$ and $-6$, what is the value of $s$?

Accepted Answer

9 For a quadratic $ax^2+bx+c=0$, the sum of roots = $-b/a$ and product = $c/a$. Here $a=1$ and $b=s$, so the sum of roots is $-s$. The given roots sum to $-3)+(-6)=-9$, therefore $-s=-9$ which gives $s=9$. Common wrong answers: $-9$ arises from forgetting the negative sign, $18$ confuses sum with product (since product = 18), and $-6$ mistakes a single root for the coefficient. Exam tip: identify $a,b,c$ first and apply "sum = -b/a, product = c/a."

Question 14

The area of a square is 15 more than 8 times its side. If the side is x, which equation is correct?

Accepted Answer

$x^2-8x-15=0$ For a square with side x the area is $x^2$. The statement says the area is 15 more than 8 times the side, so $x^2=8x+15$. Bringing all terms to one side gives $x^2-8x-15=0$, so option A is correct. Distractors fail for clear reasons: C swaps the coefficients (places 15 with 8) and thus misrepresents the given relation; B has wrong signs (+) so it does not match the sentence. Exam tip: Always translate the English sentence into a mathematical equality (area = ...) before rearranging to standard quadratic form.

Question 15

In which perfect square form can the equation $x^2+12x+36=0$ be written?

Accepted Answer

$(x+6)^2=0$ Use the identity $(x+a)^2 = x^2 + 2ax + a^2$. Here the middle coefficient 12 gives $2a=12$ so $a=6$, and $a^2=36$, hence $x^2+12x+36=(x+6)^2$. Option B, $(x-6)^2$, expands to $x^2-12x+36$ so its middle term has the opposite sign and is incorrect. Exam tip: half the middle coefficient and square it — if that equals the constant term and signs match, it's a perfect square trinomial.

Question 16

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

Accepted Answer

(4x^2-7x-4=0) Here ((2x-1)^2=4x^2-4x+1), and bringing all terms to one side gives (4x^2-7x-4=0). Apply the square identity and signs carefully.

Question 17

If $x=-4$ is a root of the equation $x^2+tx-8=0$, what is the value of $t$?

Accepted Answer

2 Substitute $x=-4$ into the equation: $(-4)^2 + t(-4) - 8 = 0$, which gives $16 - 4t - 8 = 0$. Thus $8 - 4t = 0$ and $t = 2$. The common wrong choice $-2$ typically arises from a sign error (writing $+4t$ instead of $-4t$). Exam tip: directly substitute the root and carefully handle squares and signs when simplifying.

Question 18

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

Accepted Answer

$4x^2+11x-3=0$ Expanding $(4x-1)(x+3)$ gives $4x^2+12x-x-3$. Combining like terms yields $4x^2+11x-3$, so the standard form is $4x^2+11x-3=0$. Option B has the wrong sign for the middle term (should be +11x). Options C and D result from incorrect multiplication or sign errors. Exam tip: multiply each term carefully and then combine like terms to avoid sign mistakes.

Question 19

If ((5-p)x^2+2x+9=0) is a quadratic equation, what is the correct condition for (p)?

Accepted Answer

(p\neq 5) The direct answer is B: $p\ne5$. A quadratic equation must contain a nonzero coefficient of $x^2$. In this equation that coefficient is $5-p$. Therefore we require $5-p\ne0$. Solving this condition gives $p\ne5$. Option A is wrong because if $p=5$, the $x^2$ term disappears and the equation becomes $2x+9=0$, which is linear, not quadratic. Option B is correct because every value except 5 keeps the leading coefficient nonzero. Option C, $p=0$, is unnecessary; zero is allowed and gives coefficient 5. Option D, $p\ne-5$, is not the relevant condition; $p=-5$ gives coefficient 10 and is perfectly valid. Exam cue: check the coefficient of the highest required power.

Question 20

For the equation $6x^2-13x+5=0$, if $a, b, c$ are the coefficients, what is the value of $2a+b-c$?

Accepted Answer

-6 The coefficients are $a=6,\; b=-13,\; c=5$. So $2a+b-c=2\times6+(-13)-5=12-13-5=-6$. Option B (4) is a common mistake if the sign of $c$ is handled incorrectly (adding instead of subtracting). Exam tip: always write down the values of $a,b,c$ first and track plus/minus signs carefully when substituting.

Question 21

When the equation $12-7x+x^2=0$ is written in standard form $ax^2+bx+c=0$, what is the value of $b$?

Accepted Answer

-7 Rewriting the polynomial in descending powers gives $x^2-7x+12=0$. The coefficient of $x$ is therefore $b=-7$. Option C (7) is a common mistake — it ignores the negative sign of the $x$-term. Exam tip: arrange terms in order of descending powers ($x^2, x,$ constant) to read off coefficients directly.

Question 22

Is $x=5$ a root of the equation $x^2-6x+5=0$?

Accepted Answer

Yes Substitute $x=5$: $5^2-6\cdot5+5=25-30+5=0$. Since substitution yields zero, $x=5$ is a root. Option C is incorrect because the quadratic factors as $(x-1)(x-5)=0$, so both $x=1$ and $x=5$ are roots. Exam tip: Verify a proposed root by direct substitution or factor the quadratic to find all roots.

Question 23

Write the equation $(x-6)^2 = 4x + 5$ in its standard form $ax^2+bx+c=0$.

Accepted Answer

x^2-16x+31=0 Expand the left side: $(x-6)^2 = x^2 - 12x + 36$. Move the right side $4x+5$ to the left and combine like terms: $x^2-12x+36-(4x+5)=x^2-16x+31$. Thus the standard form is $x^2-16x+31=0$. The closest distractor C ($x^2-12x+41=0$) keeps the middle term but has an incorrect constant — a typical arithmetic slip when subtracting the constant. Exam tip: always bring all terms to one side and simplify carefully, checking signs and arithmetic.

Question 24

What is the discriminant $D$ of the equation $25x^2-20x+4=0$?

Accepted Answer

0 Use the discriminant formula $D=b^2-4ac$. Here $a=25,\; b=-20,\; c=4$, so $D=(-20)^2-4\cdot25\cdot4=400-400=0$. $D=0$ means the quadratic has two real and equal roots (a repeated root). The distractor 400 is incorrect because it is just $b^2$ without subtracting $4ac$. Option 100 arises from a common arithmetic mistake in computing $4ac$, and $-20$ comes from sign or substitution errors. Exam tip: check for a perfect square trinomial — here $25x^2-20x+4=(5x-2)^2$, which immediately shows $D=0$.

Question 25

What is the nature of the roots of the equation $x^2+14x+49=0$?

Accepted Answer

Two equal real roots Compute the discriminant $\Delta=b^2-4ac$. Here $a=1,\; b=14,\; c=49$ so $\Delta=14^2-4\cdot1\cdot49=196-196=0$. When $\Delta=0$ a quadratic has two equal (repeated) real roots; using $x=-\dfrac{b}{2a}$ gives $x=-7$ as the repeated root. Why other options are wrong: A would require $\Delta>0$, C would require $\Delta<0$, and D is impossible for a quadratic (at most two roots). Exam tip: Always evaluate $\Delta$ first — it directly tells you whether roots are distinct, equal, or complex.

Introduction to Quadratic Equations

Practice questions

01 If the product of roots of (px^2+6x+9=0) is (3), what is the value of (p)?

02 In which of the following equations is (x = -2) not a root?

03 Which pair is the correct roots of \(x^2+2x-35=0\)?

04 If the roots of \(x^2+rx+36=0\) are 6 and 6, what is the value of \(r\)?

05 Which quadratic equation is obtained by simplifying \((x-3)^2+2(x-3)-8=0\)?

06 What is the standard form of \((x+2)(x-7)=3x\)?

07 If \(x^2-12x+35=0\), what is the value of the left-hand side when \(x=5\)?

08 If the roots of (x^2-14x+k=0) are equal, what is (k)?

09 In (x^2+kx+49=0), if the roots are equal and positive, which possible value of (k) is correct?

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

11 If (6x^2-18x+12=0) is divided by (6), which simplified equation is obtained?

12 Which pair of integers is useful for factoring the quadratic equation x^2 - 13x + 42 = 0?

13 If the quadratic equation \(x^2 + s x + 18 = 0\) has roots \(-3\) and \(-6\), what is the value of \(s\)?

14 The area of a square is 15 more than 8 times its side. If the side is x, which equation is correct?

15 In which perfect square form can the equation \(x^2+12x+36=0\) be written?

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

17 If \(x=-4\) is a root of the equation \(x^2+tx-8=0\), what is the value of \(t\)?

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

19 If ((5-p)x^2+2x+9=0) is a quadratic equation, what is the correct condition for (p)?

20 For the equation \(6x^2-13x+5=0\), if \(a, b, c\) are the coefficients, what is the value of \(2a+b-c\)?

21 When the equation \(12-7x+x^2=0\) is written in standard form \(ax^2+bx+c=0\), what is the value of \(b\)?

22 Is \(x=5\) a root of the equation \(x^2-6x+5=0\)?

23 Write the equation \((x-6)^2 = 4x + 5\) in its standard form \(ax^2+bx+c=0\).

24 What is the discriminant \(D\) of the equation \(25x^2-20x+4=0\)?

25 What is the nature of the roots of the equation \(x^2+14x+49=0\)?

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