Solve 89x^2-6x+4=7

$$89 x ^ { 2 } - 6 x + 4 = 7$$

Solve for x

$$x=\frac{2\sqrt{69}+3}{89}\approx 0.22037357$$

$$x=\frac{3-2\sqrt{69}}{89}\approx -0.15295784$$

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.

$$89x^{2}-6x+4=7$$

Subtract 7 from both sides of the equation.

$$89x^{2}-6x+4-7=7-7$$

Subtracting 7 from itself leaves 0.

$$89x^{2}-6x+4-7=0$$

Subtract 7 from 4.

$$89x^{2}-6x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 89 for a, -6 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 89\left(-3\right)}}{2\times 89}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 89\left(-3\right)}}{2\times 89}$$

Multiply -4 times 89.

$$x=\frac{-\left(-6\right)±\sqrt{36-356\left(-3\right)}}{2\times 89}$$

Multiply -356 times -3.

$$x=\frac{-\left(-6\right)±\sqrt{36+1068}}{2\times 89}$$

Add 36 to 1068.

$$x=\frac{-\left(-6\right)±\sqrt{1104}}{2\times 89}$$

Take the square root of 1104.

$$x=\frac{-\left(-6\right)±4\sqrt{69}}{2\times 89}$$

The opposite of -6 is 6.

$$x=\frac{6±4\sqrt{69}}{2\times 89}$$

Multiply 2 times 89.

$$x=\frac{6±4\sqrt{69}}{178}$$

Now solve the equation x=\frac{6±4\sqrt{69}}{178} when ± is plus. Add 6 to 4\sqrt{69}.

$$x=\frac{4\sqrt{69}+6}{178}$$

Divide 6+4\sqrt{69} by 178.

$$x=\frac{2\sqrt{69}+3}{89}$$

Now solve the equation x=\frac{6±4\sqrt{69}}{178} when ± is minus. Subtract 4\sqrt{69} from 6.

$$x=\frac{6-4\sqrt{69}}{178}$$

Divide 6-4\sqrt{69} by 178.

$$x=\frac{3-2\sqrt{69}}{89}$$

The equation is now solved.

$$x=\frac{2\sqrt{69}+3}{89} x=\frac{3-2\sqrt{69}}{89}$$

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.

$$89x^{2}-6x+4=7$$

Subtract 4 from both sides of the equation.

$$89x^{2}-6x+4-4=7-4$$

Subtracting 4 from itself leaves 0.

$$89x^{2}-6x=7-4$$

Subtract 4 from 7.

$$89x^{2}-6x=3$$

Divide both sides by 89.

$$\frac{89x^{2}-6x}{89}=\frac{3}{89}$$

Dividing by 89 undoes the multiplication by 89.

$$x^{2}-\frac{6}{89}x=\frac{3}{89}$$

Divide -\frac{6}{89}, the coefficient of the x term, by 2 to get -\frac{3}{89}. Then add the square of -\frac{3}{89} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{6}{89}x+\left(-\frac{3}{89}\right)^{2}=\frac{3}{89}+\left(-\frac{3}{89}\right)^{2}$$

Square -\frac{3}{89} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{6}{89}x+\frac{9}{7921}=\frac{3}{89}+\frac{9}{7921}$$

Add \frac{3}{89} to \frac{9}{7921} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{6}{89}x+\frac{9}{7921}=\frac{276}{7921}$$

Factor x^{2}-\frac{6}{89}x+\frac{9}{7921}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{89}\right)^{2}=\frac{276}{7921}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{89}\right)^{2}}=\sqrt{\frac{276}{7921}}$$

Simplify.

$$x-\frac{3}{89}=\frac{2\sqrt{69}}{89} x-\frac{3}{89}=-\frac{2\sqrt{69}}{89}$$

Add \frac{3}{89} to both sides of the equation.

$$x=\frac{2\sqrt{69}+3}{89} x=\frac{3-2\sqrt{69}}{89}$$