Solve 42x^2-696x+3240=0

$$42 { x }^{ 2 } -696x+3240=0$$

Solve for x (complex solution)

$$x=\frac{58+4\sqrt{26}i}{7}\approx 8.285714286+2.913725436i$$

$$x=\frac{-4\sqrt{26}i+58}{7}\approx 8.285714286-2.913725436i$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$42x^{2}-696x+3240=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 42 for a, -696 for b, and 3240 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-696\right)±\sqrt{\left(-696\right)^{2}-4\times 42\times 3240}}{2\times 42}$$

Square -696.

$$x=\frac{-\left(-696\right)±\sqrt{484416-4\times 42\times 3240}}{2\times 42}$$

Multiply -4 times 42.

$$x=\frac{-\left(-696\right)±\sqrt{484416-168\times 3240}}{2\times 42}$$

Multiply -168 times 3240.

$$x=\frac{-\left(-696\right)±\sqrt{484416-544320}}{2\times 42}$$

Add 484416 to -544320.

$$x=\frac{-\left(-696\right)±\sqrt{-59904}}{2\times 42}$$

Take the square root of -59904.

$$x=\frac{-\left(-696\right)±48\sqrt{26}i}{2\times 42}$$

The opposite of -696 is 696.

$$x=\frac{696±48\sqrt{26}i}{2\times 42}$$

Multiply 2 times 42.

$$x=\frac{696±48\sqrt{26}i}{84}$$

Now solve the equation x=\frac{696±48\sqrt{26}i}{84} when ± is plus. Add 696 to 48i\sqrt{26}.

$$x=\frac{696+48\sqrt{26}i}{84}$$

Divide 696+48i\sqrt{26} by 84.

$$x=\frac{58+4\sqrt{26}i}{7}$$

Now solve the equation x=\frac{696±48\sqrt{26}i}{84} when ± is minus. Subtract 48i\sqrt{26} from 696.

$$x=\frac{-48\sqrt{26}i+696}{84}$$

Divide 696-48i\sqrt{26} by 84.

$$x=\frac{-4\sqrt{26}i+58}{7}$$

The equation is now solved.

$$x=\frac{58+4\sqrt{26}i}{7} x=\frac{-4\sqrt{26}i+58}{7}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$42x^{2}-696x+3240=0$$

Subtract 3240 from both sides of the equation.

$$42x^{2}-696x+3240-3240=-3240$$

Subtracting 3240 from itself leaves 0.

$$42x^{2}-696x=-3240$$

Divide both sides by 42.

$$\frac{42x^{2}-696x}{42}=-\frac{3240}{42}$$

Dividing by 42 undoes the multiplication by 42.

$$x^{2}+\left(-\frac{696}{42}\right)x=-\frac{3240}{42}$$

Reduce the fraction \frac{-696}{42} to lowest terms by extracting and canceling out 6.

$$x^{2}-\frac{116}{7}x=-\frac{3240}{42}$$

Reduce the fraction \frac{-3240}{42} to lowest terms by extracting and canceling out 6.

$$x^{2}-\frac{116}{7}x=-\frac{540}{7}$$

Divide -\frac{116}{7}, the coefficient of the x term, by 2 to get -\frac{58}{7}. Then add the square of -\frac{58}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{116}{7}x+\left(-\frac{58}{7}\right)^{2}=-\frac{540}{7}+\left(-\frac{58}{7}\right)^{2}$$

Square -\frac{58}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{116}{7}x+\frac{3364}{49}=-\frac{540}{7}+\frac{3364}{49}$$

Add -\frac{540}{7} to \frac{3364}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{116}{7}x+\frac{3364}{49}=-\frac{416}{49}$$

Factor x^{2}-\frac{116}{7}x+\frac{3364}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{58}{7}\right)^{2}=-\frac{416}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{58}{7}\right)^{2}}=\sqrt{-\frac{416}{49}}$$

Simplify.

$$x-\frac{58}{7}=\frac{4\sqrt{26}i}{7} x-\frac{58}{7}=-\frac{4\sqrt{26}i}{7}$$

Add \frac{58}{7} to both sides of the equation.

$$x=\frac{58+4\sqrt{26}i}{7} x=\frac{-4\sqrt{26}i+58}{7}$$