Solve 7x+9/6-4x^2-9/5=3

$$\frac { 7 x + 9 } { 6 } - \frac { 4 x ^ { 2 } - 9 } { 5 } = 3$$

Solve for x

$$x = \frac{\sqrt{2089} + 35}{48} \approx 1.68136624$$

$$x=\frac{35-\sqrt{2089}}{48}\approx -0.223032907$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 30, the least common multiple of 6,5.

$$5\left(7x+9\right)-6\left(4x^{2}-9\right)=90$$

Use the distributive property to multiply 5 by 7x+9.

$$35x+45-6\left(4x^{2}-9\right)=90$$

Use the distributive property to multiply -6 by 4x^{2}-9.

$$35x+45-24x^{2}+54=90$$

Add 45 and 54 to get 99.

$$35x+99-24x^{2}=90$$

Subtract 90 from both sides.

$$35x+99-24x^{2}-90=0$$

Subtract 90 from 99 to get 9.

$$35x+9-24x^{2}=0$$

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.

$$-24x^{2}+35x+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -24 for a, 35 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-35±\sqrt{35^{2}-4\left(-24\right)\times 9}}{2\left(-24\right)}$$

Square 35.

$$x=\frac{-35±\sqrt{1225-4\left(-24\right)\times 9}}{2\left(-24\right)}$$

Multiply -4 times -24.

$$x=\frac{-35±\sqrt{1225+96\times 9}}{2\left(-24\right)}$$

Multiply 96 times 9.

$$x=\frac{-35±\sqrt{1225+864}}{2\left(-24\right)}$$

Add 1225 to 864.

$$x=\frac{-35±\sqrt{2089}}{2\left(-24\right)}$$

Multiply 2 times -24.

$$x=\frac{-35±\sqrt{2089}}{-48}$$

Now solve the equation x=\frac{-35±\sqrt{2089}}{-48} when ± is plus. Add -35 to \sqrt{2089}.

$$x=\frac{\sqrt{2089}-35}{-48}$$

Divide -35+\sqrt{2089} by -48.

$$x=\frac{35-\sqrt{2089}}{48}$$

Now solve the equation x=\frac{-35±\sqrt{2089}}{-48} when ± is minus. Subtract \sqrt{2089} from -35.

$$x=\frac{-\sqrt{2089}-35}{-48}$$

Divide -35-\sqrt{2089} by -48.

$$x=\frac{\sqrt{2089}+35}{48}$$

The equation is now solved.

$$x=\frac{35-\sqrt{2089}}{48} x=\frac{\sqrt{2089}+35}{48}$$

Steps for Completing the Square

Multiply both sides of the equation by 30, the least common multiple of 6,5.

$$5\left(7x+9\right)-6\left(4x^{2}-9\right)=90$$

Use the distributive property to multiply 5 by 7x+9.

$$35x+45-6\left(4x^{2}-9\right)=90$$

Use the distributive property to multiply -6 by 4x^{2}-9.

$$35x+45-24x^{2}+54=90$$

Add 45 and 54 to get 99.

$$35x+99-24x^{2}=90$$

Subtract 99 from both sides.

$$35x-24x^{2}=90-99$$

Subtract 99 from 90 to get -9.

$$35x-24x^{2}=-9$$

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.

$$-24x^{2}+35x=-9$$

Divide both sides by -24.

$$\frac{-24x^{2}+35x}{-24}=-\frac{9}{-24}$$

Dividing by -24 undoes the multiplication by -24.

$$x^{2}+\frac{35}{-24}x=-\frac{9}{-24}$$

Divide 35 by -24.

$$x^{2}-\frac{35}{24}x=-\frac{9}{-24}$$

Reduce the fraction \frac{-9}{-24} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{35}{24}x=\frac{3}{8}$$

Divide -\frac{35}{24}, the coefficient of the x term, by 2 to get -\frac{35}{48}. Then add the square of -\frac{35}{48} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{35}{24}x+\left(-\frac{35}{48}\right)^{2}=\frac{3}{8}+\left(-\frac{35}{48}\right)^{2}$$

Square -\frac{35}{48} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{35}{24}x+\frac{1225}{2304}=\frac{3}{8}+\frac{1225}{2304}$$

Add \frac{3}{8} to \frac{1225}{2304} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{35}{24}x+\frac{1225}{2304}=\frac{2089}{2304}$$

Factor x^{2}-\frac{35}{24}x+\frac{1225}{2304}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{35}{48}\right)^{2}=\frac{2089}{2304}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{35}{48}\right)^{2}}=\sqrt{\frac{2089}{2304}}$$

Simplify.

$$x-\frac{35}{48}=\frac{\sqrt{2089}}{48} x-\frac{35}{48}=-\frac{\sqrt{2089}}{48}$$

Add \frac{35}{48} to both sides of the equation.

$$x=\frac{\sqrt{2089}+35}{48} x=\frac{35-\sqrt{2089}}{48}$$