solving using completing the square.
xsquared+4x-10=0
You asked:
Solve the equation \({x}^{2} + 4 x - 10 = 0\) by completing the square.
MathBot Answer:
The quadratic equation \( {x}^{2} + 4 x - 10 = 0 \) has two solutions: \[ x = -2 + \sqrt{14} , \quad x = -2 - \sqrt{14} \]
The quadratic equation in standard form is: \[ x^{2} + 4 x - 10 = 0 \]
Subtract the constant term \( -10 \) from both sides of the equation: \[ x^{2} + 4 x = 10 \]
Complete the square on the left hand side by adding the square of half the linear coefficient, \( 4 \), to both sides of the equation: \[ x^{2} + 4 x + 4 = 14 \]
Rewrite the left hand side of the equation as a square: \[ \left(x + 2\right)^{2} = 14 \]
Rewrite the right hand side as a square, and subtract from both sides: \[ \left(x + 2\right)^{2} - \left(\sqrt{14}\right)^{2} = 0 \]
Use the identity \( a^2 - b^2 = (a-b)(a+b) \) to rewrite the difference of squared on the left hand side as a product: \[ \left(\left(x + 2\right) - \sqrt{14}\right) \left(\left(x + 2\right) + \sqrt{14}\right) = 0 \]
\[ x = -2 + \sqrt{14} , \quad x = -2 - \sqrt{14} \]