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Mathematics

Roots of a Quadratic Equation

द्विघात समीकरण के मूल

Roots of a Quadratic Equation, taught in Class 10 Mathematics under the chapter Quadratic Equations, introduces the values of the variable that make a quadratic expression equal to zero. Students learn to identify roots, verify them by substitution, and connect the sum and product of the roots with the coefficients. The topic also helps them form a quadratic equation when its roots are known and use these relationships to solve and check mathematical problems.

Practice questions

If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(x^2-9x+18=0\), what is the value of \((\alpha-2)(\beta-2)\)?HardLevel 32 If the roots of the quadratic equation \(5x^2+px+q=0\) are \(3\) and \(-4\), what is the value of \(\frac{q}{5}\)?HardLevel 32 If the roots of the equation \(x^2+px+q=0\) are \(5\) and \(-2\), what is the value of \(p-q\)?HardLevel 32 If c(\alphac) and c(\betac) are the roots of the equation c(x^2-6x+2=0c), what is the value of c(\alpha^2+\beta^2c)?HardLevel 32 If (\alpha) and (\beta) are roots of (x^2+2x-2=0), what is the value of (\alpha^2+\alpha\beta+\beta^2)?HardLevel 32 Which of the following is the correct pair of roots of the quadratic equation \(x^2-(2m+3)x+(m+1)(m+2)=0\)?HardLevel 32 If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(x^2-10x+24=0\), what is the value of \((\alpha-2)(\beta-2)\)?HardLevel 32 If the smaller root of (x^2-13x+42=0) is (\alpha) and the larger root is (\beta), what is (\beta-\alpha)?HardLevel 32 If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(x^2-8x+12=0\), what is the value of \(\alpha^2\beta^2\)?HardLevel 32 If α and β are roots of x^2 - 9x + 20 = 0, what can be the value of (α + β)/(α - β)?HardLevel 32 If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2-4x-5=0\), what is the value of \((\alpha+2)(\beta+2)\)?HardLevel 32 If the two roots of the equation \(x^2+kx+49=0\) are equal and negative, what is the value of \(k\)?HardLevel 32 If (\alpha) and (\beta) are roots of (x^2-14x+45=0), what is the value of (\frac{\alpha^2+\beta^2}{\alpha\beta})?HardLevel 32 Which quadratic equation has real and equal roots?HardLevel 32 If a new equation is formed by taking reciprocals of the roots of (x^2-7x+10=0), which monic equation is obtained?HardLevel 32 If (\alpha) and (\beta) are roots of (x^2-5x+5=0), what is the value of (\alpha^2+\beta^2+\alpha\beta)?HardLevel 32 If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(x^2-3x-28=0\), what is the value of \((\alpha+1)(\beta+1)\)?HardLevel 32 If (\alpha) and (\beta) are roots of (x^2-11x+24=0), what is the value of (\frac{1}{\alpha+1}+\frac{1}{\beta+1})?HardLevel 32 If the roots of (x^2-7x+k=0) are reciprocals of each other, what is the value of (k)?HardLevel 33 If the roots of (2x^2-5x+m=0) are (\alpha,\beta) and (\alpha^2+\beta^2=\frac{13}{4}), find (m).HardLevel 33