Q: Which middle step is correct while solving $9x^2-30x+8=0$ by completing the square?

$\left(x-\frac{5}{3}\right)^2=\frac{17}{9}$ First $x^2-\frac{10}{3}x+\frac{8}{9}=0$ is obtained, then $\left(x-\frac{5}{3}\right)^2=\frac{17}{9}$. In exams, divide by (a) first when $a\neq1$.

Q: What roots are obtained for $9x^2-30x+8=0$ by completing square method?

$x=\frac{5\pm\sqrt{17}}{3}$ Since $\left(x-\frac{5}{3}\right)^2=\frac{17}{9}$, $x=\frac{5\pm\sqrt{17}}{3}$. In exams, write the square root with the denominator correctly.

Question 1

What will be the roots of (14x^2-25x+6=0) by factorisation method?

Accepted Answer

(x=\frac{3}{7},1) (14x^2-25x+6=(7x-3)(2x-2)), so the roots are (\frac{3}{7}) and (1). In exams, also check by removing any common factor if present.

Question 2

What is the correct splitting of the middle term in (20x^2-43x+21=0)?

Accepted Answer

(20x^2-28x-15x+21=0) Here (ac=420) and (-28+(-15)=-43), so the correct split is (-28x-15x). In exams, even when (ac) is large, match both sum and product.

Question 3

What are the roots of (20x^2-43x+21=0)?

Accepted Answer

(x=\frac{7}{5},\frac{3}{4}) (20x^2-43x+21=(5x-7)(4x-3)), so the roots are (\frac{7}{5}) and (\frac{3}{4}). In exams, do not invert fractional roots.

Question 4

If (x^2-2(k+3)x+k^2-16=0) has equal roots, what is the value of (k)?

Accepted Answer

(k=\frac{7}{6}) For equal roots, (D=0), so (4(k+3)^2-4(k^2-16)=0) must be expanded carefully; a wrong expansion changes the answer. In exams, recheck parameter expansion.

Question 5

If the correct (k) is needed for equal roots of (x^2-2(k+3)x+k^2-16=0), which value is correct?

Accepted Answer

(k=-\frac{25}{6}) (D=4(k+3)^2-4(k^2-16)=0) gives ((k+3)^2=k^2-16), so (6k+25=0) and (k=-\frac{25}{6}). In exams, handle the constant term carefully after expansion.

Question 6

If the roots of (5x^2-(p+4)x+20=0) are (2) and (2), what is (p)?

Accepted Answer

(p=16) The sum of roots is (4), and (\frac{p+4}{5}=4), so (p=16). In exams, use (-\frac{b}{a}) for the sum.

Question 7

Which middle step is correct while solving $9x^2-30x+8=0$ by completing the square?

Accepted Answer

$\left(x-\frac{5}{3}\right)^2=\frac{17}{9}$ First $x^2-\frac{10}{3}x+\frac{8}{9}=0$ is obtained, then $\left(x-\frac{5}{3}\right)^2=\frac{17}{9}$. In exams, divide by (a) first when $a\neq1$.

Question 8

What roots are obtained for $9x^2-30x+8=0$ by completing square method?

Accepted Answer

$x=\frac{5\pm\sqrt{17}}{3}$ Since $\left(x-\frac{5}{3}\right)^2=\frac{17}{9}$, $x=\frac{5\pm\sqrt{17}}{3}$. In exams, write the square root with the denominator correctly.

Question 9

If (x^2-20x+m=0) has real and equal roots, what is the value of (m)?

Accepted Answer

(m=100) For real and equal roots, (D=0), so (400-4m=0) gives (m=100). In exams, equal roots indicate (D=0).

Question 10

Which root is common to (6x^2-19x+15=0) and (10x^2-27x+18=0)?

Accepted Answer

(x=\frac{3}{2}) The first equation has roots (\frac{3}{2},\frac{5}{3}), and the second has roots (\frac{3}{2},\frac{6}{5}). In exams, solve both equations separately for the common root.

Question 11

If one root of (x^2-21x+q=0) is (8), what will be the other root?

Accepted Answer

(13) The sum of roots is (21), so the other root is (21-8=13). In exams, use the sum when one root is given.

Question 12

If one root of (x^2-21x+q=0) is (8), what is the value of (q)?

Accepted Answer

(104) The other root is (13), so (q=8	imes13=104). In exams, when (a=1), the constant term is the product of roots.

Question 13

A student wrote (x=\frac{1}{2},-\frac{2}{3}) for (6x^2+x-2=0). How is this answer?

Accepted Answer

Correct (6x^2+x-2=(3x+2)(2x-1)), so (x=\frac{1}{2},-\frac{2}{3}) is correct. In exams, change signs carefully from factors.

Question 14

In which form can (x^2-2\sqrt{11}x+11=0) be written?

Accepted Answer

((x-\sqrt{11})^2=0) Since (11=(\sqrt{11})^2) and the middle term is (-2\sqrt{11}x), it is ((x-\sqrt{11})^2). In exams, identify perfect squares even with irrational coefficients.

Question 15

What is the root of (x^2-2\sqrt{11}x+11=0)?

Accepted Answer

(x=\sqrt{11}) Direct answer: option A, x=√11, is the repeated root. Start with x²−2√11x+11=0. Since (x−√11)²=x²−2√11x+11, the equation becomes (x−√11)²=0. A square is zero only when its base is zero, so x−√11=0 and x=√11. The root occurs twice, but the value is one repeated root. B, x=−√11, has the wrong sign; substituting it would not make the factor x−√11 zero. C, x=11, is not the square-root value and does not satisfy the equation. D, x=−11, also has the wrong sign and magnitude. Equivalently, for ax²+bx+c, the repeated-root form is (x−r)², whose middle coefficient is −2r; here r=√11. Memory cue: recognize the perfect square (x−√11)².

Question 16

What are the roots of (x^2-(r+t)x+rt=0)?

Accepted Answer

(x=r,t) The equation is equivalent to ((x-r)(x-t)=0), so the roots are (r) and (t). In exams, apply zero product rule to symbolic factors too.

Question 17

If (x^2-2cx+c^2-d^2=0), what are its roots?

Accepted Answer

(x=c+d,c-d) It is ((x-c)^2-d^2=0), so (x-c=\pm d) and (x=c\pm d). In exams, quickly recognize the difference of squares.

Question 18

After simplifying (25x^2-25(a+b)x+25ab=0), what will be the roots?

Accepted Answer

(x=a,b) Dividing the whole equation by (25) gives (x^2-(a+b)x+ab=0). In exams, removing the common factor first shortens the solution.

Question 19

If (y=x^2) is put in (x^4-17x^2+16=0), which equation is obtained?

Accepted Answer

(y^2-17y+16=0) Since (x^4=(x^2)^2=y^2), the new equation is (y^2-17y+16=0). In exams, use substitution to form a quadratic.

Question 20

What are the real solutions of (x^4-17x^2+16=0)?

Accepted Answer

(x=\pm1,\pm4) From (y^2-17y+16=0), (y=1,16), so (x^2=1,16) and (x=\pm1,\pm4). In exams, do not forget to return to (x).

Question 21

For (\frac{1}{x}+x=\frac{26}{5}), (x
eq0), what quadratic form is obtained?

Accepted Answer

(5x^2-26x+5=0) Multiplying both sides by (5x) gives (5+5x^2=26x), that is (5x^2-26x+5=0). In exams, remember the condition (x
eq0).

Question 22

What are the solutions of (\frac{1}{x}+x=\frac{26}{5}), (x
eq0)?

Accepted Answer

(x=5,\frac{1}{5}) (5x^2-26x+5=(5x-1)(x-5)), so (x=\frac{1}{5}) and (5). In exams, check whether obtained roots are valid in the original equation.

Question 23

What is the correct quadratic form of (\frac{x+4}{x}=\frac{25}{x+4}), (x
eq0,-4)?

Accepted Answer

(x^2-17x+16=0) Cross multiplication gives ((x+4)^2=25x), so (x^2+8x+16-25x=0), and (x^2-17x+16=0). In exams, cross multiply carefully.

Question 24

What are the solutions of (\frac{x+4}{x}=\frac{25}{x+4}), (x
eq0,-4)?

Accepted Answer

(x=1,16) (x^2-17x+16=(x-1)(x-16)), so (x=1) and (x=16). In exams, check solutions against excluded denominator values.

Question 25

What are the roots of (x^2-10x+7=0) by quadratic formula?

Accepted Answer

(x=5\pm3\sqrt{2}) (D=(-10)^2-4(1)(7)=72), so (x=\frac{10\pm6\sqrt{2}}{2}=5\pm3\sqrt{2}). In exams, simplify the square root.

Methods of Solving Quadratic Equations

Practice questions

01 What will be the roots of (14x^2-25x+6=0) by factorisation method?

02 What is the correct splitting of the middle term in (20x^2-43x+21=0)?

03 What are the roots of (20x^2-43x+21=0)?

04 If (x^2-2(k+3)x+k^2-16=0) has equal roots, what is the value of (k)?

05 If the correct (k) is needed for equal roots of (x^2-2(k+3)x+k^2-16=0), which value is correct?

06 If the roots of (5x^2-(p+4)x+20=0) are (2) and (2), what is (p)?

07 Which middle step is correct while solving \(9x^2-30x+8=0\) by completing the square?

08 What roots are obtained for \(9x^2-30x+8=0\) by completing square method?

09 If (x^2-20x+m=0) has real and equal roots, what is the value of (m)?

10 Which root is common to (6x^2-19x+15=0) and (10x^2-27x+18=0)?

11 If one root of (x^2-21x+q=0) is (8), what will be the other root?

12 If one root of (x^2-21x+q=0) is (8), what is the value of (q)?

13 A student wrote (x=\frac{1}{2},-\frac{2}{3}) for (6x^2+x-2=0). How is this answer?

14 In which form can (x^2-2\sqrt{11}x+11=0) be written?

15 What is the root of (x^2-2\sqrt{11}x+11=0)?

16 What are the roots of (x^2-(r+t)x+rt=0)?

17 If (x^2-2cx+c^2-d^2=0), what are its roots?

18 After simplifying (25x^2-25(a+b)x+25ab=0), what will be the roots?

19 If (y=x^2) is put in (x^4-17x^2+16=0), which equation is obtained?

20 What are the real solutions of (x^4-17x^2+16=0)?

21 For (\frac{1}{x}+x=\frac{26}{5}), (x\neq0), what quadratic form is obtained?

22 What are the solutions of (\frac{1}{x}+x=\frac{26}{5}), (x\neq0)?

23 What is the correct quadratic form of (\frac{x+4}{x}=\frac{25}{x+4}), (x\neq0,-4)?

24 What are the solutions of (\frac{x+4}{x}=\frac{25}{x+4}), (x\neq0,-4)?

25 What are the roots of (x^2-10x+7=0) by quadratic formula?

Add Muft Shiksha to your Home Screen