Q: If $\alpha,\beta$ are roots of $4x^2-13x+3=0$, what is $(\alpha-\beta)^2$?

$ \frac{121}{16}$ / $\frac{121}{16}$ $(\alpha-\beta)^2=(\alpha+\beta)^2-4\alpha\beta=\left(\frac{13}{4}\right)^2-3=\frac{121}{16}$. In exams, use this identity for the square of difference.

Q: If (x^2-10x+n=0) has two distinct real roots, which condition on (n) is correct?

(n 0), so (100-4n>0) and (n 0) with distinct real roots.

Q: Which of the following is the correct factorised form for solving the equation $5x^2-7x-6=0$?

$(5x+3)(x-2)=0$ Expanding option A gives $(5x+3)(x-2)=5x^2-10x+3x-6=5x^2-7x-6$, so it is the correct factorised form. Option B produces a middle coefficient of $+7$, while options C and D produce $+13$ and $-13$, respectively. In an exam, always expand the factors and compare the result with the original quadratic equation.

Q: What are the roots of the quadratic equation $5x^2-7x-6=0$?

$x=2,\,-\frac{3}{5}$ The quadratic factors as $5x^2-7x-6=(5x+3)(x-2)$. Thus, $(5x+3)(x-2)=0$ gives $x=-\frac{3}{5}$ or $x=2$, so option A is correct. In option B, the signs of both roots are incorrect. Exam tip: when setting each factor equal to zero, remember to change the sign while solving for the root.

Q: What are the roots of the equation $x^2+8x+5=0$?

$x=-4\pm\sqrt{11}$ Completing the square in $x^2+8x+5=0$ gives $x^2+8x+16=11$, or $(x+4)^2=11$. Hence, $x+4=\pm\sqrt{11}$, so the roots are $x=-4\pm\sqrt{11}$. Option B has the wrong sign before 4, while option D incorrectly omits the square-root sign. Exam tip: the $\pm$ symbol represents both roots, so write both values when required.

Question 1

If (\alpha,\beta) are roots of (4x^2-13x+3=0), what is (\alpha+\beta)?

Accepted Answer

( \frac{13}{4}) / (\frac{13}{4}) The sum of roots is (-\frac{b}{a}=-\frac{-13}{4}=\frac{13}{4}). In exams, keep the sign of (b) carefully.

Question 2

If (\alpha,\beta) are roots of (4x^2-13x+3=0), what is (\alpha\beta)?

Accepted Answer

( \frac{3}{4}) / (\frac{3}{4}) The product of roots is (\frac{c}{a}=\frac{3}{4}). In exams, use (\frac{c}{a}) for the product.

Question 3

If $\alpha,\beta$ are roots of $4x^2-13x+3=0$, what is $(\alpha-\beta)^2$?

Accepted Answer

$ \frac{121}{16}$ / $\frac{121}{16}$ $(\alpha-\beta)^2=(\alpha+\beta)^2-4\alpha\beta=\left(\frac{13}{4}\right)^2-3=\frac{121}{16}$. In exams, use this identity for the square of difference.

Question 4

If (x^2-10x+n=0) has no real roots, which condition on (n) is correct?

Accepted Answer

(n>25) For no real roots, (D<0), so (100-4n<0) and (n>25). In exams, connect (D<0) with no real roots.

Question 5

If (x^2-10x+n=0) has two distinct real roots, which condition on (n) is correct?

Accepted Answer

(n<25) For two distinct real roots, (D>0), so (100-4n>0) and (n<25). In exams, connect (D>0) with distinct real roots.

Question 6

Which of the following is the correct factorised form for solving the equation $5x^2-7x-6=0$?

Accepted Answer

$(5x+3)(x-2)=0$ Expanding option A gives $(5x+3)(x-2)=5x^2-10x+3x-6=5x^2-7x-6$, so it is the correct factorised form. Option B produces a middle coefficient of $+7$, while options C and D produce $+13$ and $-13$, respectively. In an exam, always expand the factors and compare the result with the original quadratic equation.

Question 7

What are the roots of the quadratic equation $5x^2-7x-6=0$?

Accepted Answer

$x=2,\,-\frac{3}{5}$ The quadratic factors as $5x^2-7x-6=(5x+3)(x-2)$. Thus, $(5x+3)(x-2)=0$ gives $x=-\frac{3}{5}$ or $x=2$, so option A is correct. In option B, the signs of both roots are incorrect. Exam tip: when setting each factor equal to zero, remember to change the sign while solving for the root.

Question 8

What are the roots of the equation $x^2+8x+5=0$?

Accepted Answer

$x=-4\pm\sqrt{11}$ Completing the square in $x^2+8x+5=0$ gives $x^2+8x+16=11$, or $(x+4)^2=11$. Hence, $x+4=\pm\sqrt{11}$, so the roots are $x=-4\pm\sqrt{11}$. Option B has the wrong sign before 4, while option D incorrectly omits the square-root sign. Exam tip: the $\pm$ symbol represents both roots, so write both values when required.

Question 9

If the roots of (x^2-6x-16=0) are (\alpha,\beta), what is (\alpha^2\beta+\alpha\beta^2)?

Accepted Answer

(-96) (\alpha^2\beta+\alpha\beta^2=\alpha\beta(\alpha+\beta)), where (\alpha+\beta=6) and (\alpha\beta=-16), so the value is (-96). In exams, factor the expression first.

Question 10

If one root of (x^2+px+25=0) is double the other and both are negative, what is (p)?

Accepted Answer

( \frac{15\sqrt{2}}{2}) / (\frac{15\sqrt{2}}{2}) Let the roots be (-r) and (-2r), then (2r^2=25) and (p=3r=\frac{15\sqrt{2}}{2}). In exams, do not forget to rationalize the denominator.

Question 11

What is the difference between the roots of (x^2-7x+3=0)?

Accepted Answer

(\sqrt{37}) Here (D=(-7)^2-4(1)(3)=37), so the difference of roots is (\frac{\sqrt{D}}{|a|}=\sqrt{37}). In exams, the difference of roots can be found directly from (D).

Question 12

What is the correct statement about real roots of (4x^2-8x+9=0)?

Accepted Answer

There are no real roots Here (D=(-8)^2-4(4)(9)=-80<0), so there are no real roots. In exams, (D<0) means no real roots.

Question 13

Which completed square form is correct for (4x^2-8x+9=0)?

Accepted Answer

(4(x-1)^2+5=0) (4x^2-8x+9=4(x-1)^2+5), so it cannot be zero for real (x). In exams, completed square form also shows the nature of roots.

Question 14

If roots of (x^2-10x+16=0) are (\alpha,\beta), which equation has roots (\alpha+3,\beta+3)?

Accepted Answer

(x^2-16x+55=0) The roots are (2,8), so new roots are (5,11), and the equation is ((x-5)(x-11)=0). In exams, form the new roots and then the new equation.

Question 15

If the roots of (x^2-14x+45=0) are (\alpha,\beta), what is (\frac{\alpha}{\beta}+\frac{\beta}{\alpha})?

Accepted Answer

( \frac{106}{45}) / (\frac{106}{45}) (\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}), where (\alpha+\beta=14) and (\alpha\beta=45), so the value is (\frac{196-90}{45}=\frac{106}{45}). In exams, convert expressions into sum and product.

Question 16

If $kx^2-14x+49=0$ has equal roots, what is the value of $k$?

Accepted Answer

1 For equal roots, the discriminant must be zero: $D=b^2-4ac=0$. Here, $a=k$, $b=-14$, and $c=49$, so $D=(-14)^2-4(k)(49)=196-196k$. Thus, $196-196k=0$, giving $k=1$. Exam tip: identify the coefficient of $x^2$ as $a=k$; the equation is quadratic only when $k\neq 0$.

Question 17

What is (k) for equal roots of (x^2-2(k+4)x+k^2=0)?

Accepted Answer

(k=-2) (D=4(k+4)^2-4k^2=0) gives ((k+4)^2=k^2), so (8k+16=0) and (k=-2). In exams, expand squares carefully.

Question 18

If ((x-4)(x-9)=14), what is the standard quadratic equation?

Accepted Answer

(x^2-13x+22=0) ((x-4)(x-9)=x^2-13x+36), so (x^2-13x+36=14) gives (x^2-13x+22=0). In exams, bring all terms to one side after expansion.

Question 19

What are the roots of (x^2-13x+22=0) by quadratic formula?

Accepted Answer

(x=\frac{13\pm9}{2}) Here (D=(-13)^2-4(1)(22)=81), so (x=\frac{13\pm9}{2}). In exams, if (D) is a perfect square, the answer simplifies quickly.

Methods of Solving Quadratic Equations

Practice questions

01 If (\alpha,\beta) are roots of (4x^2-13x+3=0), what is (\alpha+\beta)?

02 If (\alpha,\beta) are roots of (4x^2-13x+3=0), what is (\alpha\beta)?

03 If \(\alpha,\beta\) are roots of \(4x^2-13x+3=0\), what is \((\alpha-\beta)^2\)?

04 If (x^2-10x+n=0) has no real roots, which condition on (n) is correct?

05 If (x^2-10x+n=0) has two distinct real roots, which condition on (n) is correct?

06 Which of the following is the correct factorised form for solving the equation \(5x^2-7x-6=0\)?

07 What are the roots of the quadratic equation \(5x^2-7x-6=0\)?

08 What are the roots of the equation \(x^2+8x+5=0\)?

09 If the roots of (x^2-6x-16=0) are (\alpha,\beta), what is (\alpha^2\beta+\alpha\beta^2)?

10 If one root of (x^2+px+25=0) is double the other and both are negative, what is (p)?

11 What is the difference between the roots of (x^2-7x+3=0)?

12 What is the correct statement about real roots of (4x^2-8x+9=0)?

13 Which completed square form is correct for (4x^2-8x+9=0)?

14 If roots of (x^2-10x+16=0) are (\alpha,\beta), which equation has roots (\alpha+3,\beta+3)?

15 If the roots of (x^2-14x+45=0) are (\alpha,\beta), what is (\frac{\alpha}{\beta}+\frac{\beta}{\alpha})?

16 If \(kx^2-14x+49=0\) has equal roots, what is the value of \(k\)?

17 What is (k) for equal roots of (x^2-2(k+4)x+k^2=0)?

18 If ((x-4)(x-9)=14), what is the standard quadratic equation?

19 What are the roots of (x^2-13x+22=0) by quadratic formula?

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