Q: Which middle step is correct while solving $5x^2-18x+9=0$ by completing the square?

$\left(x-\frac{9}{5}\right)^2=\frac{36}{25}$ First $x^2-\frac{18}{5}x+\frac{9}{5}=0$ is obtained, then $\left(x-\frac{9}{5}\right)^2=\frac{36}{25}$. In exams, divide by (a) first when $a\neq1$.

Q: If (x^2-14x+m=0) has real and equal roots, what is the value of (m)?

(m=49) The direct answer is option A: $m=49$. For a quadratic $ax^2+bx+c=0$, real and equal roots occur exactly when the discriminant $D=b^2-4ac$ is zero. Here $a=1$, $b=-14$, and $c=m$. Thus $D=(-14)^2-4(1)(m)=196-4m$. Set it equal to zero: $196-4m=0$, so $4m=196$, and $m=49$. Option A is correct. Option B, $m=14$, gives $D=140$, so there are two distinct real roots. Option C, $m=28$, gives $D=84$, also two distinct real roots. Option D, $m=196$, gives $D=-588$, so the roots are not real. The key fact is: equal real roots mean the discriminant is exactly zero, not merely positive or negative.

Q: If one root of (x^2-9x+q=0) is (4), what is the value of (q)?

(20) The other root is (5), so (q=4\times5=20). In exams, (c) equals the product of roots when (a=1).

Question 1

What is the correct splitting of the middle term in (15x^2+16x+4=0)?

Accepted Answer

(15x^2+10x+6x+4=0) Here (ac=60) and (10+6=16), so (16x) is split as (10x+6x). In exams, check both sum (b) and product (ac).

Question 2

What are the roots of (15x^2+16x+4=0)?

Accepted Answer

(x=-\frac{2}{3},-\frac{2}{5}) (15x^2+16x+4=(3x+2)(5x+2)), so the roots are (-\frac{2}{3}) and (-\frac{2}{5}). In exams, write fractional roots in simplest form.

Question 3

If (x^2-2(k-1)x+k^2-4=0) has equal roots, what is the value of (k)?

Accepted Answer

(k=\frac{5}{2}) For equal roots, (D=0), so (4(k-1)^2-4(k^2-4)=0) gives (-2k+5=0). In exams, expand (D) carefully in parameter questions.

Question 4

If the roots of (3x^2-(p+2)x+12=0) are (2) and (2), what is (p)?

Accepted Answer

(p=10) The sum of roots is (4), and (\frac{p+2}{3}=4), so (p=10). In exams, find the sum of roots using (-\frac{b}{a}).

Question 5

Which middle step is correct while solving $5x^2-18x+9=0$ by completing the square?

Accepted Answer

$\left(x-\frac{9}{5}\right)^2=\frac{36}{25}$ First $x^2-\frac{18}{5}x+\frac{9}{5}=0$ is obtained, then $\left(x-\frac{9}{5}\right)^2=\frac{36}{25}$. In exams, divide by (a) first when $a\neq1$.

Question 6

What will be the roots of (5x^2-18x+9=0)?

Accepted Answer

(x=3,\frac{3}{5}) (5x^2-18x+9=(5x-3)(x-3)), so the roots are (\frac{3}{5}) and (3). In exams, verify the answer quickly by factorisation.

Question 7

If (x^2-14x+m=0) has real and equal roots, what is the value of (m)?

Accepted Answer

(m=49) The direct answer is option A: $m=49$. For a quadratic $ax^2+bx+c=0$, real and equal roots occur exactly when the discriminant $D=b^2-4ac$ is zero. Here $a=1$, $b=-14$, and $c=m$. Thus $D=(-14)^2-4(1)(m)=196-4m$. Set it equal to zero: $196-4m=0$, so $4m=196$, and $m=49$. Option A is correct. Option B, $m=14$, gives $D=140$, so there are two distinct real roots. Option C, $m=28$, gives $D=84$, also two distinct real roots. Option D, $m=196$, gives $D=-588$, so the roots are not real. The key fact is: equal real roots mean the discriminant is exactly zero, not merely positive or negative.

Question 8

Which root is common to (3x^2-10x+8=0) and (4x^2-12x+8=0)?

Accepted Answer

(x=2) The roots of the first equation are (2,\frac{4}{3}), and the roots of the second are (2,1). In exams, solve both equations separately for the common root.

Question 9

If one root of (x^2-9x+q=0) is (4), what will be the other root?

Accepted Answer

(5) The sum of roots is (9), so the other root is (9-4=5). In exams, use the sum when one root is given.

Question 10

If one root of (x^2-9x+q=0) is (4), what is the value of (q)?

Accepted Answer

(20) The other root is (5), so (q=4	imes5=20). In exams, (c) equals the product of roots when (a=1).

Question 11

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

Accepted Answer

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

Question 12

In which form can (x^2-2\sqrt{5}x+5=0) be written?

Accepted Answer

((x-\sqrt{5})^2=0) Since (5=(\sqrt{5})^2) and the middle term is (-2\sqrt{5}x), it is ((x-\sqrt{5})^2). In exams, identify perfect squares even with irrational coefficients.

Question 13

What are the roots of (x^2-(m+n)x+mn=0)?

Accepted Answer

(x=m,n) The equation is equivalent to ((x-m)(x-n)=0), so the roots are (m) and (n). In exams, the same rule applies to symbolic factors.

Question 14

If (x^2-2px+p^2-q^2=0), what are its roots?

Accepted Answer

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

Question 15

After simplifying (9x^2-9(r+s)x+9rs=0), what will be the roots?

Accepted Answer

(x=r,s) Dividing the whole equation by (9) gives (x^2-(r+s)x+rs=0). In exams, removing the common factor first makes solving easier.

Question 16

If (y=x^2) is put in (x^4-10x^2+9=0), which equation is obtained?

Accepted Answer

(y^2-10y+9=0) Since (x^4=(x^2)^2=y^2), the new equation is (y^2-10y+9=0). In exams, substitution can simplify a difficult form.

Question 17

What are the real solutions of (x^4-10x^2+9=0)?

Accepted Answer

(x=\pm1,\pm3) From (y^2-10y+9=0), (y=1,9), so (x^2=1,9) and (x=\pm1,\pm3). In exams, do not forget to return to (x).

Question 18

For (\frac{1}{x}+x=\frac{10}{3}), (x
eq0), what quadratic form is obtained?

Accepted Answer

(3x^2-10x+3=0) Multiplying both sides by (3x) gives (3+3x^2=10x), that is (3x^2-10x+3=0). In exams, remember the condition (x
eq0).

Question 19

What are the solutions of (\frac{1}{x}+x=\frac{10}{3}), (x
eq0)?

Accepted Answer

(x=3,\frac{1}{3}) (3x^2-10x+3=(3x-1)(x-3)), so (x=\frac{1}{3}) and (3). In exams, check whether obtained roots are valid in the original equation.

Question 20

What is the correct quadratic form of (\frac{x+2}{x}=\frac{9}{x+2}), (x
eq0,-2)?

Accepted Answer

(x^2-5x+4=0) Cross multiplication gives ((x+2)^2=9x), so (x^2+4x+4-9x=0), and (x^2-5x+4=0). In exams, cross multiply carefully.

Question 21

What are the solutions of (\frac{x+2}{x}=\frac{9}{x+2}), (x
eq0,-2)?

Accepted Answer

(x=1,4) (x^2-5x+4=(x-1)(x-4)), so (x=1) and (x=4). In exams, check solutions against excluded denominator values.

Question 22

What are the roots of (x^2-6x+2=0) by quadratic formula?

Accepted Answer

(x=3\pm\sqrt{7}) (D=(-6)^2-4(1)(2)=28), so (x=\frac{6\pm2\sqrt{7}}{2}=3\pm\sqrt{7}). In exams, simplify the square root.

Question 23

If the sum of roots of (4x^2+px+16=0) is (-6), what is (p)?

Accepted Answer

(24) The sum of roots is (-\frac{p}{4}), so (-\frac{p}{4}=-6) gives (p=24). In exams, remember the sum formula (-\frac{b}{a}).

Question 24

If the product of roots of (5x^2-11x+p=0) is (\frac{2}{5}), what is (p)?

Accepted Answer

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

Question 25

If the roots of (x^2-13x+40=0) are (\alpha) and (\beta), what is (\alpha^2+\beta^2)?

Accepted Answer

(89) (\alpha+\beta=13) and (\alpha\beta=40), so (\alpha^2+\beta^2=13^2-2(40)=89). In exams, remember ((\alpha+\beta)^2-2\alpha\beta).

Methods of Solving Quadratic Equations

Practice questions

01 What is the correct splitting of the middle term in (15x^2+16x+4=0)?

02 What are the roots of (15x^2+16x+4=0)?

03 If (x^2-2(k-1)x+k^2-4=0) has equal roots, what is the value of (k)?

04 If the roots of (3x^2-(p+2)x+12=0) are (2) and (2), what is (p)?

05 Which middle step is correct while solving \(5x^2-18x+9=0\) by completing the square?

06 What will be the roots of (5x^2-18x+9=0)?

07 If (x^2-14x+m=0) has real and equal roots, what is the value of (m)?

08 Which root is common to (3x^2-10x+8=0) and (4x^2-12x+8=0)?

09 If one root of (x^2-9x+q=0) is (4), what will be the other root?

10 If one root of (x^2-9x+q=0) is (4), what is the value of (q)?

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

12 In which form can (x^2-2\sqrt{5}x+5=0) be written?

13 What are the roots of (x^2-(m+n)x+mn=0)?

14 If (x^2-2px+p^2-q^2=0), what are its roots?

15 After simplifying (9x^2-9(r+s)x+9rs=0), what will be the roots?

16 If (y=x^2) is put in (x^4-10x^2+9=0), which equation is obtained?

17 What are the real solutions of (x^4-10x^2+9=0)?

18 For (\frac{1}{x}+x=\frac{10}{3}), (x\neq0), what quadratic form is obtained?

19 What are the solutions of (\frac{1}{x}+x=\frac{10}{3}), (x\neq0)?

20 What is the correct quadratic form of (\frac{x+2}{x}=\frac{9}{x+2}), (x\neq0,-2)?

21 What are the solutions of (\frac{x+2}{x}=\frac{9}{x+2}), (x\neq0,-2)?

22 What are the roots of (x^2-6x+2=0) by quadratic formula?

23 If the sum of roots of (4x^2+px+16=0) is (-6), what is (p)?

24 If the product of roots of (5x^2-11x+p=0) is (\frac{2}{5}), what is (p)?

25 If the roots of (x^2-13x+40=0) are (\alpha) and (\beta), what is (\alpha^2+\beta^2)?

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