Q: If $\alpha,\beta$ are roots of $2x^2-9x+4=0$, what is $(\alpha-\beta)^2$?

$ \frac{65}{4}$ / $\frac{65}{4}$ $(\alpha-\beta)^2=(\alpha+\beta)^2-4\alpha\beta=\left(\frac{9}{2}\right)^2-8=\frac{65}{4}$. In exams, use this identity for square of difference.

Q: If (x^2-4x+n=0) has no real roots, which condition on (n) is correct?

(n>4) The direct answer is option A: $n>4$. For a quadratic equation $ax^2+bx+c=0$, the discriminant is $D=b^2-4ac$. It has no real roots when $D 4$. Thus option A is correct. Option B, $n<4$, gives a positive discriminant and therefore two real roots. Option C, $n=4$, gives $D=0$, so there is one repeated real root, not no real roots. Option D, $n\le4$, incorrectly includes values below 4 and also includes 4; those cases have real roots. The important boundary is 4. Memory cue: for no real roots, always write $D<0$, and reverse the inequality when dividing by a negative number.

Q: If (x^2-4x+n=0) has two distinct real roots, which condition on (n) is correct?

(n 0), so (16-4n>0) and (n 0) with distinct roots.

Q: What are the roots of the equation $x^2+4x+1=0$?

$x=-2\pm\sqrt{3}$ Rewrite the equation by completing the square: $x^2+4x+1=0\Rightarrow x^2+4x+4=3\Rightarrow (x+2)^2=3$. Thus, $x+2=\pm\sqrt{3}$, giving $x=-2\pm\sqrt{3}$. Option B has the wrong sign for the constant term in the solution. In an exam, remember that $\pm$ represents both roots.

Question 1

If the roots of (x^2-11x+30=0) are (\alpha) and (\beta), what is (\alpha^2+\beta^2)?

Accepted Answer

(61) (\alpha+\beta=11) and (\alpha\beta=30), so (\alpha^2+\beta^2=(11)^2-2(30)=61). In exams, remember the identity ((\alpha+\beta)^2-2\alpha\beta).

Question 2

If the roots of (x^2-7x+10=0) are (\alpha) and (\beta), what is (\frac{1}{\alpha}+\frac{1}{\beta})?

Accepted Answer

( \frac{7}{10}) / (\frac{7}{10}) (\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}=\frac{7}{10}). In exams, first write sum and product for reciprocal questions.

Question 3

If (\alpha,\beta) are roots of (2x^2-9x+4=0), what is (\alpha+\beta)?

Accepted Answer

( \frac{9}{2}) / (\frac{9}{2}) The sum of roots is (-\frac{b}{a}=-\frac{-9}{2}=\frac{9}{2}). In exams, keep the sign of (b) carefully.

Question 4

If (\alpha,\beta) are roots of (2x^2-9x+4=0), what is (\alpha\beta)?

Accepted Answer

(2) The product of roots is (\frac{c}{a}=\frac{4}{2}=2). In exams, use (\frac{c}{a}) for the product.

Question 5

If $\alpha,\beta$ are roots of $2x^2-9x+4=0$, what is $(\alpha-\beta)^2$?

Accepted Answer

$ \frac{65}{4}$ / $\frac{65}{4}$ $(\alpha-\beta)^2=(\alpha+\beta)^2-4\alpha\beta=\left(\frac{9}{2}\right)^2-8=\frac{65}{4}$. In exams, use this identity for square of difference.

Question 6

If (x^2-4x+n=0) has no real roots, which condition on (n) is correct?

Accepted Answer

(n>4) The direct answer is option A: $n>4$. For a quadratic equation $ax^2+bx+c=0$, the discriminant is $D=b^2-4ac$. It has no real roots when $D<0$. Here $a=1$, $b=-4$, and $c=n$. Therefore $D=(-4)^2-4(1)(n)=16-4n$. Require $16-4n<0$. Subtract 16: $-4n<-16$. Dividing by $-4$ reverses the inequality, giving $n>4$. Thus option A is correct. Option B, $n<4$, gives a positive discriminant and therefore two real roots. Option C, $n=4$, gives $D=0$, so there is one repeated real root, not no real roots. Option D, $n\le4$, incorrectly includes values below 4 and also includes 4; those cases have real roots. The important boundary is 4. Memory cue: for no real roots, always write $D<0$, and reverse the inequality when dividing by a negative number.

Question 7

If (x^2-4x+n=0) has two distinct real roots, which condition on (n) is correct?

Accepted Answer

(n<4) For two distinct real roots, (D>0), so (16-4n>0) and (n<4). In exams, connect (D>0) with distinct roots.

Question 8

Which of the following is the correct factorised form for solving the equation 2x² − 3x − 2 = 0?

Accepted Answer

(2x+1)(x−2)=0 Expanding (2x+1)(x−2) gives 2x² − 4x + x − 2 = 2x² − 3x − 2, so option A is correct. The closest distractor is option B, whose expansion is 2x² + 3x − 2; the sign of the coefficient of x is wrong. In an exam, expand the factors and compare the result with the original equation.

Question 9

What are the roots of the equation $x^2+4x+1=0$?

Accepted Answer

$x=-2\pm\sqrt{3}$ Rewrite the equation by completing the square: $x^2+4x+1=0\Rightarrow x^2+4x+4=3\Rightarrow (x+2)^2=3$. Thus, $x+2=\pm\sqrt{3}$, giving $x=-2\pm\sqrt{3}$. Option B has the wrong sign for the constant term in the solution. In an exam, remember that $\pm$ represents both roots.

Question 10

If the roots of (x^2-2x-8=0) are (\alpha,\beta), what is (\alpha^2\beta+\alpha\beta^2)?

Accepted Answer

(-16) (\alpha^2\beta+\alpha\beta^2=\alpha\beta(\alpha+\beta)), where (\alpha+\beta=2) and (\alpha\beta=-8), so the value is (-16). In exams, factor the expression first.

Question 11

If one root of (x^2+px+9=0) is double the other and both are negative, what is (p)?

Accepted Answer

(3\sqrt{2}) Let the roots be (-r) and (-2r), then (2r^2=9) and the sum is (-3r), so (p=3r=\frac{9}{\sqrt{2}}). In exams, assume the roots and form equations carefully.

Question 12

If one root of (x^2+px+9=0) is double the other and both are negative, what is the correct simplified value of (p)?

Accepted Answer

( \frac{9\sqrt{2}}{2}) / (\frac{9\sqrt{2}}{2}) (\frac{9}{\sqrt{2}}) simplifies to (\frac{9\sqrt{2}}{2}). In exams, do not forget to rationalize the denominator.

Question 13

What is the difference between the roots of (x^2-3x+1=0)?

Accepted Answer

(\sqrt{5}) The square of the difference is (D/a^2=5), so the difference is (\sqrt{5}). In exams, the difference of roots is (\frac{\sqrt{D}}{|a|}).

Question 14

What is the correct statement about real roots of (2x^2-4x+5=0)?

Accepted Answer

There are no real roots Here (D=(-4)^2-4(2)(5)=-24<0), so there are no real roots. In exams, (D<0) means no real roots.

Question 15

Which completed square form is correct for (2x^2-4x+5=0)?

Accepted Answer

(2(x-1)^2+3=0) (2x^2-4x+5=2(x-1)^2+3), so it cannot be zero for real (x). In exams, completed square form also shows no real roots.

Question 16

If roots of (x^2-6x+5=0) are (\alpha,\beta), which equation has roots (\alpha+1,\beta+1)?

Accepted Answer

(x^2-8x+12=0) The roots are (1,5), so new roots are (2,6), and the equation is ((x-2)(x-6)=0). In exams, form the new roots and then the new equation.

Question 17

If the roots of (x^2-8x+15=0) are (\alpha,\beta), what is (\frac{\alpha}{\beta}+\frac{\beta}{\alpha})?

Accepted Answer

( \frac{34}{15}) / (\frac{34}{15}) (\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}), where (\alpha+\beta=8) and (\alpha\beta=15), so the value is (\frac{64-30}{15}=\frac{34}{15}). In exams, convert expressions into sum and product.

Question 18

If the quadratic equation $kx^2-6x+9=0$ 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, $a=k$, $b=-6$, and $c=9$, so $D=(-6)^2-4(k)(9)=36-36k$. Thus, $36-36k=0$, giving $k=1$. Exam tip: identify $a$ as $k$, since it is the coefficient of $x^2$.

Question 19

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

Accepted Answer

(k=-1) (D=4(k+2)^2-4k^2=0) gives ((k+2)^2=k^2), so (4k+4=0) and (k=-1). In exams, expand squares carefully.

Question 20

If ((x-2)(x-5)=m) and (m=0), what are the solutions?

Accepted Answer

(x=2,5) Putting (m=0) gives ((x-2)(x-5)=0), so (x=2) or (x=5). In exams, apply zero product rule directly.

Question 21

If ((x-2)(x-5)=6), what is the standard quadratic equation?

Accepted Answer

(x^2-7x+4=0) ((x-2)(x-5)=x^2-7x+10), so (x^2-7x+10=6) gives (x^2-7x+4=0). In exams, bring all terms to one side after expansion.

Question 22

What are the roots of (x^2-7x+4=0) by quadratic formula?

Accepted Answer

(x=\frac{7\pm\sqrt{33}}{2}) Here (D=(-7)^2-4(1)(4)=33), so (x=\frac{7\pm\sqrt{33}}{2}). In exams, finding (D) correctly is important.

Question 23

If (2x^2+8x+1=0) is solved by completing the square, which middle step is correct?

Accepted Answer

((x+2)^2=\frac{7}{2}) First (x^2+4x+\frac{1}{2}=0) is obtained, then adding (4) gives ((x+2)^2=\frac{7}{2}). In exams, divide by (a) first when (a
eq1).

Question 24

What roots are obtained for (2x^2+8x+1=0) by completing the square method?

Accepted Answer

(x=-2\pm\frac{\sqrt{14}}{2}) Since ((x+2)^2=\frac{7}{2}), (x=-2\pm\sqrt{\frac{7}{2}}=-2\pm\frac{\sqrt{14}}{2}). In exams, write the square root in simplified form.

Question 25

What will be the roots of (8x^2-14x+3=0) by factorisation method?

Accepted Answer

(x=\frac{1}{4},\frac{3}{2}) (8x^2-14x+3=(4x-1)(2x-3)), so the roots are (\frac{1}{4}) and (\frac{3}{2}). In exams, set each linear factor equal to zero.

Methods of Solving Quadratic Equations

Practice questions

01 If the roots of (x^2-11x+30=0) are (\alpha) and (\beta), what is (\alpha^2+\beta^2)?

02 If the roots of (x^2-7x+10=0) are (\alpha) and (\beta), what is (\frac{1}{\alpha}+\frac{1}{\beta})?

03 If (\alpha,\beta) are roots of (2x^2-9x+4=0), what is (\alpha+\beta)?

04 If (\alpha,\beta) are roots of (2x^2-9x+4=0), what is (\alpha\beta)?

05 If \(\alpha,\beta\) are roots of \(2x^2-9x+4=0\), what is \((\alpha-\beta)^2\)?

06 If (x^2-4x+n=0) has no real roots, which condition on (n) is correct?

07 If (x^2-4x+n=0) has two distinct real roots, which condition on (n) is correct?

08 Which of the following is the correct factorised form for solving the equation 2x² − 3x − 2 = 0?

09 What are the roots of the equation \(x^2+4x+1=0\)?

10 If the roots of (x^2-2x-8=0) are (\alpha,\beta), what is (\alpha^2\beta+\alpha\beta^2)?

11 If one root of (x^2+px+9=0) is double the other and both are negative, what is (p)?

12 If one root of (x^2+px+9=0) is double the other and both are negative, what is the correct simplified value of (p)?

13 What is the difference between the roots of (x^2-3x+1=0)?

14 What is the correct statement about real roots of (2x^2-4x+5=0)?

15 Which completed square form is correct for (2x^2-4x+5=0)?

16 If roots of (x^2-6x+5=0) are (\alpha,\beta), which equation has roots (\alpha+1,\beta+1)?

17 If the roots of (x^2-8x+15=0) are (\alpha,\beta), what is (\frac{\alpha}{\beta}+\frac{\beta}{\alpha})?

18 If the quadratic equation \(kx^2-6x+9=0\) has equal roots, what is the value of \(k\)?

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

20 If ((x-2)(x-5)=m) and (m=0), what are the solutions?

21 If ((x-2)(x-5)=6), what is the standard quadratic equation?

22 What are the roots of (x^2-7x+4=0) by quadratic formula?

23 If (2x^2+8x+1=0) is solved by completing the square, which middle step is correct?

24 What roots are obtained for (2x^2+8x+1=0) by completing the square method?

25 What will be the roots of (8x^2-14x+3=0) by factorisation method?

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