The equations $x^2 - ax + b = 0$ and $x^2 + bx - a = 0$ have a common root, then

Let $\alpha$ be a common root of the given equations. $\therefore \, a^2 - a\alpha + b = 0$ and $a^2 + b\alpha - a= 0$ . $ \Rightarrow \, (a + b) \alpha - (a + b) = 0 $ $\Rightarrow \, (a + b) (\alpha - 1) = 0$ $\Rightarrow \, a + b = 0$ or $\alpha = 1$

The equations $x^2 - ax + b = 0$ and $x^2 + bx

Top Questions on Quadratic Equations

If the roots of the quadratic equation $ x^2 - 7x + 12 = 0 $ are $ \alpha $ and $ \beta $, then the value of $ \alpha + \beta $ is:
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For $ X = (x_1, x_2, x_3)^T \in \mathbb{R}^3 $, consider the quadratic form:

\[ Q(X) = 2x_1^2 + 2x_2^2 + 3x_3^2 + 4x_1x_2 + 2x_1x_3 + 2x_2x_3. \] Let $ M $ be the symmetric matrix associated with the quadratic form $ Q(X) $ with respect to the standard basis of $ \mathbb{R}^3 $.
Let $ Y = (y_1, y_2, y_3)^T \in \mathbb{R}^3 $ be a non-zero vector, and let

\[ a_n = \frac{Y^T(M + I_3)^{n+1}Y}{Y^T(M + I_3)^n Y}, \quad n = 1, 2, 3, \dots \] Then, the value of $ \lim_{n \to \infty} a_n $ is equal to (in integer).
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If $ \alpha $ and $ \beta $ are negative real roots of the quadratic equation $ x^2 - (p + 2)x + (2p + 9) = 0 $ and $ p \in (\alpha, \beta) $. Then the value of $ \beta^2 - 2\alpha $ is:
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The sum of the squares of all the roots of the equation $ x^2 + [2x - 3] - 4 = 0 $ is:
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Let $ \alpha, \beta $ be the roots of the equation $ x^2 - ax - b = 0 $ with $ \text{Im}(\alpha) <\text{Im(\beta) $. Let $ P_n = \alpha^n - \beta^n $. If} \[ P_3 = -5\sqrt{7}, \, P_4 = -3\sqrt{7}, \, P_5 = 11\sqrt{7}, \, P_6 = 45\sqrt{7}, \] then $ |\alpha^4 + \beta^4| $ is equal to:
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Notes on Quadratic Equations

Concepts Used:

Quadratic Equations

A polynomial that has two roots or is of degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b, and c are the real numbers.

Consider the following equation ax²+bx+c=0, where a≠0 and a, b, and c are real coefficients.

The solution of a quadratic equation can be found using the formula, x=((-b±√(b²-4ac))/2a)

Two important points to keep in mind are:

A polynomial equation has at least one root.
A polynomial equation of degree ‘n’ has ‘n’ roots.

There are basically four methods of solving quadratic equations. They are:

Factoring
Completing the square
Using Quadratic Formula
Taking the square root