What are the roots of (x^2-13x+22=0) by quadratic formula?
Correct answer: A
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.
View question detailsMathematics
Methods of Solving Quadratic Equations
द्विघात समीकरणों को हल करने की विधियाँ
In this Class 10 Mathematics topic from the chapter Quadratic Equations, students learn how to find the values of an unknown variable that satisfy a quadratic equation. They practise solving equations by factorisation, completing the square, and using the quadratic formula, while learning when each method is useful. The topic also develops skills in identifying coefficients, calculating the discriminant, checking solutions, and interpreting whether an equation has two, one, or no real roots.
TOPIC PRACTICE
Quiz this set
Up to 20 questions from this page. Select your focus, then start.
20 questions
Choose questions
Hard · Level 36 · quadratic,quadratic-formula,application
View options
- (x=\frac{13\pm9}{2})
- (x=\frac{-13\pm9}{2})
- (x=13\pm9)
- (x=\frac{13\pm\sqrt{22}}{2})
Expert · Level 34 · quadratic,expert,factorisation
View options
- (x=\frac{3}{7},1)
- (x=-\frac{3}{7},-1)
- (x=\frac{7}{3},1)
- (x=6,\frac{1}{14})
Expert · Level 34 · quadratic,middle-term-splitting,ac-method
View options
- (20x^2-28x-15x+21=0)
- (20x^2-35x-8x+21=0)
- (20x^2-20x-23x+21=0)
- (20x^2-30x-13x+21=0)
Expert · Level 34 · quadratic,roots,factorisation
View options
- (x=\frac{7}{5},\frac{3}{4})
- (x=-\frac{7}{5},-\frac{3}{4})
- (x=\frac{5}{7},\frac{4}{3})
- (x=7,3)
Expert · Level 34 · quadratic,discriminant,parameter
View options
- (k=\frac{7}{6})
- (k=-\frac{7}{6})
- (k=3)
- (k=16)
Expert · Level 34 · quadratic,discriminant,correction
View options
- (k=\frac{7}{6})
- (k=-\frac{25}{6})
- (k=\frac{25}{6})
- (k=-\frac{7}{6})
Expert · Level 34 · quadratic,roots-parameter,sum-of-roots
View options
- (p=16)
- (p=20)
- (p=12)
- (p=8)
Expert · Level 34 · quadratic,completing-square,expert
View options
- \(\left(x-\frac{5}{3}\right)^2=\frac{17}{9}\)
- \(\left(x+\frac{5}{3}\right)^2=\frac{17}{9}\)
- \(\left(x-\frac{10}{3}\right)^2=\frac{8}{9}\)
- \(\left(x-\frac{5}{3}\right)^2=\frac{25}{9}\)
Expert · Level 34 · quadratic,completing-square,irrational-roots
View options
- \(x=\frac{5\pm\sqrt{17}}{3}\)
- \(x=\frac{-5\pm\sqrt{17}}{3}\)
- \(x=5\pm\sqrt{17}\)
- \(x=\frac{10\pm\sqrt{17}}{9}\)
Expert · Level 34 · quadratic,discriminant,equal-roots
View options
- (m=100)
- (m=20)
- (m=40)
- (m=400)
Expert · Level 34 · quadratic,common-root,factorisation
View options
- (x=\frac{3}{2})
- (x=\frac{5}{3})
- (x=\frac{6}{5})
- (x=2)
Expert · Level 34 · quadratic,roots,sum-of-roots
View options
- (13)
- (8)
- (21)
- (-13)
Expert · Level 34 · quadratic,parameter,product-of-roots
View options
- (104)
- (168)
- (13)
- (21)
Expert · Level 34 · quadratic,error-checking,factorisation
View options
- Correct
- Incorrect because the second root is (\frac{2}{3})
- Incorrect because the first root is (-\frac{1}{2})
- Incorrect because both roots are positive
Expert · Level 34 · quadratic,perfect-square,irrational
View options
- ((x-\sqrt{11})^2=0)
- ((x+\sqrt{11})^2=0)
- ((x-11)^2=0)
- ((x-2\sqrt{11})^2=0)
Expert · Level 34 · quadratic,repeated-root,irrational
View options
- (x=\sqrt{11})
- (x=-\sqrt{11})
- (x=11)
- (x=-11)
Expert · Level 34 · quadratic,symbolic,factorisation
View options
- (x=r,t)
- (x=-r,-t)
- (x=r+t,rt)
- (x=r-t,t-r)
Expert · Level 34 · quadratic,symbolic,difference-of-squares
View options
- (x=c+d,c-d)
- (x=-c+d,-c-d)
- (x=d+c,d-c)
- (x=c^2,d^2)
Expert · Level 34 · quadratic,simplification,symbolic
View options
- (x=a,b)
- (x=-a,-b)
- (x=25a,25b)
- (x=a+b,ab)
Expert · Level 34 · quadratic,substitution,biquadratic
View options
- (y^2-17y+16=0)
- (y^2+17y+16=0)
- (y^2-17x+16=0)
- (y-17y+16=0)
What will be the roots of (14x^2-25x+6=0) by factorisation method?
Correct answer: A
(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.
View question detailsWhat is the correct splitting of the middle term in (20x^2-43x+21=0)?
Correct answer: A
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.
View question detailsWhat are the roots of (20x^2-43x+21=0)?
Correct answer: A
(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.
View question detailsIf (x^2-2(k+3)x+k^2-16=0) has equal roots, what is the value of (k)?
Correct answer: A
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.
View question detailsIf the correct (k) is needed for equal roots of (x^2-2(k+3)x+k^2-16=0), which value is correct?
Correct answer: B
(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.
View question detailsIf the roots of (5x^2-(p+4)x+20=0) are (2) and (2), what is (p)?
Correct answer: A
The sum of roots is (4), and (\frac{p+4}{5}=4), so (p=16). In exams, use (-\frac{b}{a}) for the sum.
View question detailsWhich middle step is correct while solving \(9x^2-30x+8=0\) by completing the square?
Correct answer: A
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 \(a\neq1\).
View question detailsWhat roots are obtained for \(9x^2-30x+8=0\) by completing square method?
Correct answer: A
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.
View question detailsIf (x^2-20x+m=0) has real and equal roots, what is the value of (m)?
Correct answer: A
For real and equal roots, (D=0), so (400-4m=0) gives (m=100). In exams, equal roots indicate (D=0).
View question detailsWhich root is common to (6x^2-19x+15=0) and (10x^2-27x+18=0)?
Correct answer: A
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.
View question detailsIf one root of (x^2-21x+q=0) is (8), what will be the other root?
Correct answer: A
The sum of roots is (21), so the other root is (21-8=13). In exams, use the sum when one root is given.
View question detailsIf one root of (x^2-21x+q=0) is (8), what is the value of (q)?
Correct answer: A
The other root is (13), so (q=8\times13=104). In exams, when (a=1), the constant term is the product of roots.
View question detailsA student wrote (x=\frac{1}{2},-\frac{2}{3}) for (6x^2+x-2=0). How is this answer?
Correct answer: A
(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.
View question detailsIn which form can (x^2-2\sqrt{11}x+11=0) be written?
Correct answer: A
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.
View question detailsWhat is the root of (x^2-2\sqrt{11}x+11=0)?
Correct answer: A
((x-\sqrt{11})^2=0), so the repeated root is (\sqrt{11}). In exams, ((x-a)^2=0) gives (x=a).
View question detailsWhat are the roots of (x^2-(r+t)x+rt=0)?
Correct answer: A
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.
View question detailsIf (x^2-2cx+c^2-d^2=0), what are its roots?
Correct answer: A
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.
View question detailsAfter simplifying (25x^2-25(a+b)x+25ab=0), what will be the roots?
Correct answer: A
Dividing the whole equation by (25) gives (x^2-(a+b)x+ab=0). In exams, removing the common factor first shortens the solution.
View question detailsIf (y=x^2) is put in (x^4-17x^2+16=0), which equation is obtained?
Correct answer: A
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.
View question detailsQUIZ COMPLETE