For (\frac{1}{x}+x=\frac{50}{7}), (x\neq0), what quadratic form is obtained?
Correct answer: A
Multiplying both sides by (7x) gives (7+7x^2=50x), that is (7x^2-50x+7=0). In exams, remember the condition (x\neq0).
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.
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Expert · Level 36 · quadratic,reciprocal-equation,standard-form
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- (7x^2-50x+7=0)
- (7x^2+50x+7=0)
- (x^2-50x+7=0)
- (7x^2-50=0)
Expert · Level 36 · quadratic,reciprocal-equation,solutions
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- (x=7,\frac{1}{7})
- (x=-7,-\frac{1}{7})
- (x=50,7)
- (x=\frac{50}{7},1)
Expert · Level 36 · quadratic,rational-equation,standard-form
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- (x^2-37x+36=0)
- (x^2+12x-13=0)
- (x^2-49x+36=0)
- (x^2+37x+36=0)
Expert · Level 36 · quadratic,rational-equation,solutions
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- (x=1,36)
- (x=-1,-36)
- (x=6,49)
- (x=37,36)
Expert · Level 36 · quadratic,quadratic-formula,perfect-square-discriminant
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- (x=1,13)
- (x=-1,-13)
- (x=7\pm6)
- (x=14\pm\sqrt{13})
Expert · Level 36 · quadratic,sum-of-roots,parameter
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- (90)
- (-90)
- (10)
- (324)
Expert · Level 36 · quadratic,product-of-roots,parameter
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- (4)
- (10)
- ( \frac{1}{25}) / (\frac{1}{25})
- ( \frac{23}{5}) / (\frac{23}{5})
Expert · Level 36 · quadratic,roots-expression,expert
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- (369)
- (729)
- (180)
- (549)
Expert · Level 36 · quadratic,roots-expression,reciprocal
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- ( \frac{8}{45}) / (\frac{8}{45})
- ( \frac{45}{8}) / (\frac{45}{8})
- (24)
- (135)
Expert · Level 36 · quadratic,sum-of-roots,formula
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- ( \frac{31}{8}) / (\frac{31}{8})
- ( \frac{15}{8}) / (\frac{15}{8})
- ( -\frac{31}{8}) / (-\frac{31}{8})
- (8)
Expert · Level 36 · quadratic,product-of-roots,formula
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- ( \frac{15}{8}) / (\frac{15}{8})
- ( \frac{31}{8}) / (\frac{31}{8})
- (15)
- (-\frac{15}{8})
Expert · Level 36 · quadratic,roots-expression,identity
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- \( \frac{481}{64}\) / \(\frac{481}{64}\)
- \( \frac{961}{64}\) / \(\frac{961}{64}\)
- \( \frac{15}{2}\) / \(\frac{15}{2}\)
- \( \frac{225}{64}\) / \(\frac{225}{64}\)
Expert · Level 36 · quadratic,discriminant,inequality
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- (n>64)
- (n<64)
- (n=64)
- (n\le64)
Expert · Level 36 · quadratic,discriminant,distinct-roots
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- (n<64)
- (n>64)
- (n=64)
- (n\ge64)
Expert · Level 36 · quadratic,factorisation,audit
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- ((8x+5)(x-3)=0)
- ((8x-5)(x+3)=0)
- ((x+5)(8x-3)=0)
- ((8x-3)(x+5)=0)
Expert · Level 36 · quadratic,factorisation,error-checking
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- ((8x+5)(x-3)=0)
- ((8x-5)(x+3)=0)
- ((x+5)(8x-3)=0)
- ((8x-3)(x+5)=0)
Expert · Level 36 · quadratic,factorisation,verification
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- ((4x+3)(2x-5)=0)
- ((4x-3)(2x+5)=0)
- ((8x+5)(x-3)=0)
- ((x+5)(8x-3)=0)
Expert · Level 36 · quadratic,roots,factorisation
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- (x=\frac{5}{2},-\frac{3}{4})
- (x=-\frac{5}{2},\frac{3}{4})
- (x=2,-5)
- (x=\frac{14}{8},-1)
Expert · Level 36 · quadratic,completing-square,irrational-roots
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- ((x+7)^2=39)
- ((x-7)^2=39)
- ((x+14)^2=10)
- ((x+7)^2=10)
Expert · Level 36 · quadratic equations,completing the square,roots,algebra
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- \(x=-7\pm\sqrt{39}\)
- \(x=7\pm\sqrt{39}\)
- \(x=-14\pm\sqrt{39}\)
- \(x=-7\pm39\)
What are the solutions of (\frac{1}{x}+x=\frac{50}{7}), (x\neq0)?
Correct answer: A
(7x^2-50x+7=(7x-1)(x-7)), so (x=\frac{1}{7}) and (7). In exams, check whether obtained roots are valid in the original equation.
View question detailsWhat is the correct quadratic form of (\frac{x+6}{x}=\frac{49}{x+6}), (x\neq0,-6)?
Correct answer: A
Cross multiplication gives ((x+6)^2=49x), so (x^2+12x+36-49x=0), and (x^2-37x+36=0). In exams, cross multiply carefully.
View question detailsWhat are the solutions of (\frac{x+6}{x}=\frac{49}{x+6}), (x\neq0,-6)?
Correct answer: A
(x^2-37x+36=(x-1)(x-36)), so (x=1) and (x=36). In exams, check solutions against excluded denominator values.
View question detailsWhat are the roots of (x^2-14x+13=0) by quadratic formula?
Correct answer: A
(D=(-14)^2-4(1)(13)=144), so (x=\frac{14\pm12}{2}) gives (1) and (13). In exams, if (D) is a perfect square, simplify quickly.
View question detailsIf the sum of roots of (9x^2+px+36=0) is (-10), what is (p)?
Correct answer: A
The sum of roots is (-\frac{p}{9}), so (-\frac{p}{9}=-10) gives (p=90). In exams, remember the sum formula (-\frac{b}{a}).
View question detailsIf the product of roots of (10x^2-23x+p=0) is (\frac{2}{5}), what is (p)?
Correct answer: A
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}).
View question detailsIf the roots of (x^2-27x+180=0) are (\alpha) and (\beta), what is (\alpha^2+\beta^2)?
Correct answer: A
(\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).
View question detailsIf the roots of (x^2-24x+135=0) are (\alpha) and (\beta), what is (\frac{1}{\alpha}+\frac{1}{\beta})?
Correct answer: A
(\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.
View question detailsIf (\alpha,\beta) are roots of (8x^2-31x+15=0), what is (\alpha+\beta)?
Correct answer: A
The sum of roots is (-\frac{b}{a}=-\frac{-31}{8}=\frac{31}{8}). In exams, keep the sign of (b) carefully.
View question detailsIf (\alpha,\beta) are roots of (8x^2-31x+15=0), what is (\alpha\beta)?
Correct answer: A
The product of roots is (\frac{c}{a}=\frac{15}{8}). In exams, use (\frac{c}{a}) for the product.
View question detailsIf \(\alpha,\beta\) are roots of \(8x^2-31x+15=0\), what is \((\alpha-\beta)^2\)?
Correct answer: A
\((\alpha-\beta)^2=(\alpha+\beta)^2-4\alpha\beta=\left(\frac{31}{8}\right)^2-\frac{15}{2}=\frac{481}{64}\). In exams, convert fractions to a common denominator.
View question detailsIf (x^2-16x+n=0) has no real roots, which condition on (n) is correct?
Correct answer: A
For no real roots, (D<0), so (256-4n<0) and (n>64). In exams, connect (D<0) with no real roots.
View question detailsIf (x^2-16x+n=0) has two distinct real roots, which condition on (n) is correct?
Correct answer: A
For two distinct real roots, (D>0), so (256-4n>0) and (n<64). In exams, connect (D>0) with distinct real roots.
View question detailsWhich factorised form is correct for solving (8x^2-23x-15=0)?
Correct answer: A
((8x+5)(x-3)=8x^2-19x-15), so it is not for the given equation. In exams, verify each option by expansion.
View question detailsWhat is the correct factorised form of (8x^2-23x-15=0)?
Correct answer: C
((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.
View question detailsWhich factorised form is correct for solving (8x^2-14x-15=0)?
Correct answer: A
((4x+3)(2x-5)=8x^2-20x+6x-15=8x^2-14x-15), so it is correct. In exams, verify factorisation by expanding.
View question detailsWhat are the roots of (8x^2-14x-15=0)?
Correct answer: A
((4x+3)(2x-5)=0), so (x=-\frac{3}{4}) and (\frac{5}{2}). In exams, change signs while writing roots.
View question detailsIf (x^2+14x+10=0), which form is correct by completing square?
Correct answer: A
Adding (49) to (x^2+14x=-10) gives ((x+7)^2=39). In exams, add the same number to both sides.
View question detailsWhat are the roots of the equation \(x^2+14x+10=0\)?
Correct answer: A
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.
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