Q: To factorise the quadratic equation $x^2-17x+72=0$, which two numbers should have a sum of 17 and a product of 72?

8 and 9 The numbers 8 and 9 have sum $8+9=17$ and product $8\times9=72$. Hence, $x^2-17x+72=(x-8)(x-9)$. Although 6 and 12 have product 72, their sum is 18, so they do not work. Exam tip: For $x^2+bx+c$, look for two numbers whose sum is $b$ and whose product is $c$.

Q: If the roots of the equation $x^2+vx+28=0$ are $-4$ and $-7$, what is the value of $v$?

11 For the quadratic equation $x^2+vx+28=0$, the sum of the roots is $-v$. The given roots have sum $-4+(-7)=-11$. Hence, $-v=-11$, so $v=11$. Option $-11$ is the sum of the roots, not the value of $v$. Exam tip: For $ax^2+bx+c=0$, the sum of roots is $-b/a$.

Q: The area of a square is 21 more than 10 times its side. If the side of the square is $x$, which quadratic equation is obtained?

$x^2-10x-21=0$ If the side of the square is $x$, its area is $x^2$. The statement says that the area is 21 more than $10x$, so $x^2=10x+21$. Moving all terms to one side gives $x^2-10x-21=0$. In option B, the sign of the $10x$ term is incorrect. Exam tip: Translate “more than” by adding to the stated quantity.

Q: Which of the following is the perfect-square form of the equation $x^2-18x+81=0$?

$(x-9)^2=0$ $(x-9)^2=x^2-2\times9\times x+9^2=x^2-18x+81$. Hence, the given equation is equivalent to $(x-9)^2=0$. In contrast, $(x+9)^2$ has the middle term $+18x$, so it is not correct. Exam tip: use the identity $x^2-2ax+a^2=(x-a)^2$.

Q: Which equation is obtained when $(3x+4)^2=7x+16$ is written in standard quadratic form?

$9x^2+17x=0$ $(3x+4)^2=9x^2+24x+16$. Therefore, $9x^2+24x+16=7x+16$. Moving all terms to the left gives $9x^2+24x+16-7x-16=0$, so $9x^2+17x=0$. Option B incorrectly treats $24x-7x$ as $31x$. Exam tip: write every quadratic equation in the form $ax^2+bx+c=0$ and combine like terms carefully.

Q: If 4 is a root of the equation $x^2+sx-32=0$, what is the value of $s$?

4 A root makes the equation equal to zero. Substituting $x=4$ gives $4^2+4s-32=0$, or $16+4s-32=0$. Hence, $4s=16$, so $s=4$. The value $-4$ does not satisfy the equation for $x=4$. Exam tip: Substitute the given root directly into the equation to find an unknown coefficient.

Question 1

In which option is the constant term absent but the equation is quadratic?

Accepted Answer

(2x^2+7x=0) In (2x^2+7x=0), the (x^2) term is present and the constant term is absent. An equation can be quadratic even without a constant term.

Question 2

If (8x^2-32x+24=0) is divided by (8), which simplified equation is obtained?

Accepted Answer

(x^2-4x+3=0) Dividing every term by (8) gives (x^2-4x+3=0). Dividing by a common nonzero factor does not change the roots.

Question 3

To factorise the quadratic equation $x^2-17x+72=0$, which two numbers should have a sum of 17 and a product of 72?

Accepted Answer

8 and 9 The numbers 8 and 9 have sum $8+9=17$ and product $8\times9=72$. Hence, $x^2-17x+72=(x-8)(x-9)$. Although 6 and 12 have product 72, their sum is 18, so they do not work. Exam tip: For $x^2+bx+c$, look for two numbers whose sum is $b$ and whose product is $c$.

Question 4

If the roots of the equation $x^2+vx+28=0$ are $-4$ and $-7$, what is the value of $v$?

Accepted Answer

11 For the quadratic equation $x^2+vx+28=0$, the sum of the roots is $-v$. The given roots have sum $-4+(-7)=-11$. Hence, $-v=-11$, so $v=11$. Option $-11$ is the sum of the roots, not the value of $v$. Exam tip: For $ax^2+bx+c=0$, the sum of roots is $-b/a$.

Question 5

The area of a square is 21 more than 10 times its side. If the side of the square is $x$, which quadratic equation is obtained?

Accepted Answer

$x^2-10x-21=0$ If the side of the square is $x$, its area is $x^2$. The statement says that the area is 21 more than $10x$, so $x^2=10x+21$. Moving all terms to one side gives $x^2-10x-21=0$. In option B, the sign of the $10x$ term is incorrect. Exam tip: Translate “more than” by adding to the stated quantity.

Question 6

Which of the following is the perfect-square form of the equation $x^2-18x+81=0$?

Accepted Answer

$(x-9)^2=0$ $(x-9)^2=x^2-2\times9\times x+9^2=x^2-18x+81$. Hence, the given equation is equivalent to $(x-9)^2=0$. In contrast, $(x+9)^2$ has the middle term $+18x$, so it is not correct. Exam tip: use the identity $x^2-2ax+a^2=(x-a)^2$.

Question 7

Which equation is obtained when $(3x+4)^2=7x+16$ is written in standard quadratic form?

Accepted Answer

$9x^2+17x=0$ $(3x+4)^2=9x^2+24x+16$. Therefore, $9x^2+24x+16=7x+16$. Moving all terms to the left gives $9x^2+24x+16-7x-16=0$, so $9x^2+17x=0$. Option B incorrectly treats $24x-7x$ as $31x$. Exam tip: write every quadratic equation in the form $ax^2+bx+c=0$ and combine like terms carefully.

Question 8

If 4 is a root of the equation $x^2+sx-32=0$, what is the value of $s$?

Accepted Answer

4 A root makes the equation equal to zero. Substituting $x=4$ gives $4^2+4s-32=0$, or $16+4s-32=0$. Hence, $4s=16$, so $s=4$. The value $-4$ does not satisfy the equation for $x=4$. Exam tip: Substitute the given root directly into the equation to find an unknown coefficient.

Question 9

What is the value of (b^2+ac) in (7x^2-5x+2=0)?

Accepted Answer

(39) Here (a=7), (b=-5), (c=2), so (b^2+ac=25+14=39). In (b^2), the negative sign becomes positive after squaring.

Question 10

Which equation will have sum of roots (0)?

Accepted Answer

(x^2-36=0) In (x^2-36=0), (b=0), so the sum of roots is (-\frac{b}{a}=0). If the (x) term is absent, the sum can be (0).

Question 11

The base and height of a right triangle are $x$ cm and $x+2$ cm, respectively. If its area is $24\text{ cm}^2$, which quadratic equation in $x$ is correct?

Accepted Answer

$x^2+2x-48=0$ The area of a right triangle is $\frac{1}{2}\times\text{base}\times\text{height}$. Therefore, $\frac{1}{2}x(x+2)=24$. Multiplying both sides by 2 gives $x(x+2)=48$. On expanding, we get $x^2+2x-48=0$, so option A is correct. Option B incorrectly misses the step of multiplying by 2. Exam tip: In area-based questions, carefully account for the $\frac{1}{2}$ in the triangle-area formula.

Question 12

If ((n² − 16)x² − 3x + 7 = 0) is a quadratic equation, what is the correct condition on n?

Accepted Answer

n ≠ ±4 A quadratic equation in x must have a non-zero coefficient of x². In the given equation, that coefficient is n² − 16. Therefore, the necessary condition is n² − 16 ≠ 0. Solving the equality that must be avoided gives n² = 16, so n = 4 or n = −4. Hence both values must be excluded, which can be written compactly as n ≠ ±4. Option A excludes only n = 4 and still permits −4, while option B excludes only −4 and still permits 4; both are incomplete. If option D were used, the x² coefficient would become zero and the equation would reduce to −3x + 7 = 0, which is linear. Thus option C is the only complete and correct answer.

Introduction to Quadratic Equations

Practice questions

01 In which option is the constant term absent but the equation is quadratic?

02 If (8x^2-32x+24=0) is divided by (8), which simplified equation is obtained?

03 To factorise the quadratic equation \(x^2-17x+72=0\), which two numbers should have a sum of 17 and a product of 72?

04 If the roots of the equation \(x^2+vx+28=0\) are \(-4\) and \(-7\), what is the value of \(v\)?

05 The area of a square is 21 more than 10 times its side. If the side of the square is \(x\), which quadratic equation is obtained?

06 Which of the following is the perfect-square form of the equation \(x^2-18x+81=0\)?

07 Which equation is obtained when \((3x+4)^2=7x+16\) is written in standard quadratic form?

08 If 4 is a root of the equation \(x^2+sx-32=0\), what is the value of \(s\)?

09 What is the value of (b^2+ac) in (7x^2-5x+2=0)?

10 Which equation will have sum of roots (0)?

11 The base and height of a right triangle are \(x\) cm and \(x+2\) cm, respectively. If its area is \(24\text{ cm}^2\), which quadratic equation in \(x\) is correct?

12 If ((n² − 16)x² − 3x + 7 = 0) is a quadratic equation, what is the correct condition on n?

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