Q: If $\alpha,\beta$ are roots of $8x^2-31x+15=0$, what is $(\alpha-\beta)^2$?

$ \frac{481}{64}$ / $\frac{481}{64}$ $(\alpha-\beta)^2=(\alpha+\beta)^2-4\alpha\beta=\left(\frac{31}{8}\right)^2-\frac{15}{2}=\frac{481}{64}$. In exams, convert fractions to a common denominator.

Q: If (x^2-16x+n=0) has two distinct real roots, which condition on (n) is correct?

(n 0), so (256-4n>0) and (n 0) with distinct real roots.

Question 1

If the product of roots of (10x^2-23x+p=0) is (\frac{2}{5}), what is (p)?

Accepted Answer

(4) The product of roots is (\frac{p}{10}), so (\frac{p}{10}=\frac{2}{5}) gives (p=4). In exams, use the product formula (\frac{c}{a}).

Question 2

If the roots of (x^2-27x+180=0) are (\alpha) and (\beta), what is (\alpha^2+\beta^2)?

Accepted Answer

(369) (\alpha+\beta=27) and (\alpha\beta=180), so (\alpha^2+\beta^2=27^2-2(180)=369). In exams, remember ((\alpha+\beta)^2-2\alpha\beta).

Question 3

If the roots of (x^2-24x+135=0) are (\alpha) and (\beta), what is (\frac{1}{\alpha}+\frac{1}{\beta})?

Accepted Answer

( \frac{8}{45}) / (\frac{8}{45}) (\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}=\frac{24}{135}=\frac{8}{45}). In exams, write the answer in simplest form.

Question 4

If (\alpha,\beta) are roots of (8x^2-31x+15=0), what is (\alpha+\beta)?

Accepted Answer

( \frac{31}{8}) / (\frac{31}{8}) The sum of roots is (-\frac{b}{a}=-\frac{-31}{8}=\frac{31}{8}). In exams, keep the sign of (b) carefully.

Question 5

If (\alpha,\beta) are roots of (8x^2-31x+15=0), what is (\alpha\beta)?

Accepted Answer

( \frac{15}{8}) / (\frac{15}{8}) The product of roots is (\frac{c}{a}=\frac{15}{8}). In exams, use (\frac{c}{a}) for the product.

Question 6

If $\alpha,\beta$ are roots of $8x^2-31x+15=0$, what is $(\alpha-\beta)^2$?

Accepted Answer

$ \frac{481}{64}$ / $\frac{481}{64}$ $(\alpha-\beta)^2=(\alpha+\beta)^2-4\alpha\beta=\left(\frac{31}{8}\right)^2-\frac{15}{2}=\frac{481}{64}$. In exams, convert fractions to a common denominator.

Question 7

If (x^2-16x+n=0) has no real roots, which condition on (n) is correct?

Accepted Answer

(n>64) For no real roots, (D<0), so (256-4n<0) and (n>64). In exams, connect (D<0) with no real roots.

Question 8

If (x^2-16x+n=0) has two distinct real roots, which condition on (n) is correct?

Accepted Answer

(n<64) For two distinct real roots, (D>0), so (256-4n>0) and (n<64). In exams, connect (D>0) with distinct real roots.

Question 9

Which factorised form is correct for solving (8x^2-23x-15=0)?

Accepted Answer

((8x+5)(x-3)=0) ((8x+5)(x-3)=8x^2-19x-15), so it is not for the given equation. In exams, verify each option by expansion.

Question 10

What is the correct factorised form of (8x^2-23x-15=0)?

Accepted Answer

((x+5)(8x-3)=0) ((x+5)(8x-3)) does not expand to the given equation, so the options must be checked carefully. The correct factorisation is not present among careless options.

Question 11

Which factorised form is correct for solving (8x^2-14x-15=0)?

Accepted Answer

((4x+3)(2x-5)=0) ((4x+3)(2x-5)=8x^2-20x+6x-15=8x^2-14x-15), so it is correct. In exams, verify factorisation by expanding.

Question 12

What are the roots of (8x^2-14x-15=0)?

Accepted Answer

(x=\frac{5}{2},-\frac{3}{4}) ((4x+3)(2x-5)=0), so (x=-\frac{3}{4}) and (\frac{5}{2}). In exams, change signs while writing roots.

Question 13

If (x^2+14x+10=0), which form is correct by completing square?

Accepted Answer

((x+7)^2=39) The direct answer is A, (x+7)^2=39. Begin with x^2+14x+10=0. Move the constant term to the other side: x^2+14x=-10. Half of the coefficient of x is 7, and its square is 49. Add 49 to both sides: x^2+14x+49=-10+49. The left side becomes (x+7)^2, so the result is (x+7)^2=39. Option A is therefore correct. Option B has x-7; expanding it gives x^2-14x+49, so its middle sign is wrong. Option C uses (x+14)^2, whose expansion has 28x, not 14x, and it also keeps the wrong right side. Option D has the correct square form but the right side is incorrectly written as 10 instead of 39. Completing the square always requires adding the same number to both sides. Memory cue: half the linear coefficient, square it, add it to both sides.

Question 14

What are the roots of the equation $x^2+14x+10=0$?

Accepted Answer

$x=-7\pm\sqrt{39}$ Rewrite the quadratic by completing the square: $x^2+14x+10=(x+7)^2-39$. Thus, $(x+7)^2=39$, so $x+7=\pm\sqrt{39}$ and $x=-7\pm\sqrt{39}$. Option B has the wrong sign for the constant term in the root expression, while option D omits the square root. In an exam, remember that $\pm$ represents both roots.

Question 15

If the roots of (x^2-12x-28=0) are (\alpha,\beta), what is (\alpha^2\beta+\alpha\beta^2)?

Accepted Answer

(-336) (\alpha^2\beta+\alpha\beta^2=\alpha\beta(\alpha+\beta)), where (\alpha+\beta=12) and (\alpha\beta=-28), so the value is (-336). In exams, factor the expression first.

Question 16

If one root of (x^2+px+64=0) is double the other and both are negative, what is (p)?

Accepted Answer

(12\sqrt{2}) Let the roots be (-r) and (-2r), then (2r^2=64) gives (r=4\sqrt{2}), and (p=3r=12\sqrt{2}). In exams, keep signs of both roots carefully.

Question 17

What is the difference between the roots of (x^2-13x+7=0)?

Accepted Answer

(\sqrt{141}) Here (D=(-13)^2-4(1)(7)=141), so the difference of roots is (\frac{\sqrt{D}}{|a|}=\sqrt{141}). In exams, the difference of roots can be found directly from (D).

Question 18

What is the correct statement about real roots of (7x^2-14x+19=0)?

Accepted Answer

There are no real roots Here (D=(-14)^2-4(7)(19)=-336<0), so there are no real roots. In exams, (D<0) means no real roots.

Question 19

Which completed square form is correct for (7x^2-14x+19=0)?

Accepted Answer

(7(x-1)^2+12=0) (7x^2-14x+19=7(x-1)^2+12), so it cannot be zero for real (x). In exams, completed square form also shows the nature of roots.

Question 20

If roots of (x^2-16x+48=0) are (\alpha,\beta), which equation has roots (\alpha+6,\beta+6)?

Accepted Answer

(x^2-28x+180=0) The roots are (4,12), so new roots are (10,18), and the equation is ((x-10)(x-18)=0). In exams, form the new roots and then the new equation.

Question 21

If the roots of (x^2-26x+165=0) are (\alpha,\beta), what is (\frac{\alpha}{\beta}+\frac{\beta}{\alpha})?

Accepted Answer

( \frac{346}{165}) / (\frac{346}{165}) (\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}), where (\alpha+\beta=26) and (\alpha\beta=165), so the value is (\frac{676-330}{165}=\frac{346}{165}). In exams, convert expressions into sum and product.

Question 22

If the quadratic equation $kx^2-26x+169=0$ 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, $a=k$, $b=-26$, and $c=169$. Thus, $D=(-26)^2-4(k)(169)=676-676k=0$, giving $k=1$. Values such as $13$ or $169$ do not make the discriminant zero. Exam tip: For equal roots in a parameter-based quadratic equation, immediately apply $b^2-4ac=0$.

Question 23

What is (k) for equal roots of (x^2-2(k+7)x+k^2=0)?

Accepted Answer

(k=-\frac{7}{2}) (D=4(k+7)^2-4k^2=0) gives ((k+7)^2=k^2), so (14k+49=0) and (k=-\frac{7}{2}). In exams, expand squares carefully.

Question 24

If ((x-7)(x-15)=26), what is the standard quadratic equation?

Accepted Answer

(x^2-22x+79=0) ((x-7)(x-15)=x^2-22x+105), so (x^2-22x+105=26) gives (x^2-22x+79=0). In exams, bring all terms to one side after expansion.

Question 25

What are the roots of (x^2-22x+79=0) by quadratic formula?

Accepted Answer

(x=11\pm\sqrt{42}) Here (D=(-22)^2-4(1)(79)=168), so (x=\frac{22\pm2\sqrt{42}}{2}=11\pm\sqrt{42}). In exams, simplify (D) correctly.

Methods of Solving Quadratic Equations

Practice questions

01 If the product of roots of (10x^2-23x+p=0) is (\frac{2}{5}), what is (p)?

02 If the roots of (x^2-27x+180=0) are (\alpha) and (\beta), what is (\alpha^2+\beta^2)?

03 If the roots of (x^2-24x+135=0) are (\alpha) and (\beta), what is (\frac{1}{\alpha}+\frac{1}{\beta})?

04 If (\alpha,\beta) are roots of (8x^2-31x+15=0), what is (\alpha+\beta)?

05 If (\alpha,\beta) are roots of (8x^2-31x+15=0), what is (\alpha\beta)?

06 If \(\alpha,\beta\) are roots of \(8x^2-31x+15=0\), what is \((\alpha-\beta)^2\)?

07 If (x^2-16x+n=0) has no real roots, which condition on (n) is correct?

08 If (x^2-16x+n=0) has two distinct real roots, which condition on (n) is correct?

09 Which factorised form is correct for solving (8x^2-23x-15=0)?

10 What is the correct factorised form of (8x^2-23x-15=0)?

11 Which factorised form is correct for solving (8x^2-14x-15=0)?

12 What are the roots of (8x^2-14x-15=0)?

13 If (x^2+14x+10=0), which form is correct by completing square?

14 What are the roots of the equation \(x^2+14x+10=0\)?

15 If the roots of (x^2-12x-28=0) are (\alpha,\beta), what is (\alpha^2\beta+\alpha\beta^2)?

16 If one root of (x^2+px+64=0) is double the other and both are negative, what is (p)?

17 What is the difference between the roots of (x^2-13x+7=0)?

18 What is the correct statement about real roots of (7x^2-14x+19=0)?

19 Which completed square form is correct for (7x^2-14x+19=0)?

20 If roots of (x^2-16x+48=0) are (\alpha,\beta), which equation has roots (\alpha+6,\beta+6)?

21 If the roots of (x^2-26x+165=0) are (\alpha,\beta), what is (\frac{\alpha}{\beta}+\frac{\beta}{\alpha})?

22 If the quadratic equation \(kx^2-26x+169=0\) has equal roots, what is the value of \(k\)?

23 What is (k) for equal roots of (x^2-2(k+7)x+k^2=0)?

24 If ((x-7)(x-15)=26), what is the standard quadratic equation?

25 What are the roots of (x^2-22x+79=0) by quadratic formula?

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