Solve 6x+0.5*15x^2=18

$$6x+0.5 \times 15 { x }^{ 2 } =18$$

Solve for x

$$x=1.2$$

$$x=-2$$

Steps Using the Quadratic Formula

Multiply 0.5 and 15 to get 7.5.

$$6x+7.5x^{2}=18$$

Subtract 18 from both sides.

$$6x+7.5x^{2}-18=0$$

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.

$$7.5x^{2}+6x-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7.5 for a, 6 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times 7.5\left(-18\right)}}{2\times 7.5}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\times 7.5\left(-18\right)}}{2\times 7.5}$$

Multiply -4 times 7.5.

$$x=\frac{-6±\sqrt{36-30\left(-18\right)}}{2\times 7.5}$$

Multiply -30 times -18.

$$x=\frac{-6±\sqrt{36+540}}{2\times 7.5}$$

Add 36 to 540.

$$x=\frac{-6±\sqrt{576}}{2\times 7.5}$$

Take the square root of 576.

$$x=\frac{-6±24}{2\times 7.5}$$

Multiply 2 times 7.5.

$$x=\frac{-6±24}{15}$$

Now solve the equation x=\frac{-6±24}{15} when ± is plus. Add -6 to 24.

$$x=\frac{18}{15}$$

Reduce the fraction \frac{18}{15} to lowest terms by extracting and canceling out 3.

$$x=\frac{6}{5}$$

Now solve the equation x=\frac{-6±24}{15} when ± is minus. Subtract 24 from -6.

$$x=-\frac{30}{15}$$

Divide -30 by 15.

$$x=-2$$

The equation is now solved.

$$x=\frac{6}{5} x=-2$$

Steps for Completing the Square

Multiply 0.5 and 15 to get 7.5.

$$6x+7.5x^{2}=18$$

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.

$$7.5x^{2}+6x=18$$

Divide both sides of the equation by 7.5, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{7.5x^{2}+6x}{7.5}=\frac{18}{7.5}$$

Dividing by 7.5 undoes the multiplication by 7.5.

$$x^{2}+\frac{6}{7.5}x=\frac{18}{7.5}$$

Divide 6 by 7.5 by multiplying 6 by the reciprocal of 7.5.

$$x^{2}+0.8x=\frac{18}{7.5}$$

Divide 18 by 7.5 by multiplying 18 by the reciprocal of 7.5.

$$x^{2}+0.8x=2.4$$

Divide 0.8, the coefficient of the x term, by 2 to get 0.4. Then add the square of 0.4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+0.8x+0.4^{2}=2.4+0.4^{2}$$

Square 0.4 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+0.8x+0.16=2.4+0.16$$

Add 2.4 to 0.16 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+0.8x+0.16=2.56$$

Factor x^{2}+0.8x+0.16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+0.4\right)^{2}=2.56$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+0.4\right)^{2}}=\sqrt{2.56}$$

Simplify.

$$x+0.4=\frac{8}{5} x+0.4=-\frac{8}{5}$$

Subtract 0.4 from both sides of the equation.

$$x=\frac{6}{5} x=-2$$