Solve 1.5*(1-x)*(1+x)+7x-3=0

$$1.5 \times (1-x) \times (1+x)+7x-3=0$$

Solve for x

$$x = \frac{2 \sqrt{10} + 7}{3} \approx 4.44151844$$

$$x=\frac{7-2\sqrt{10}}{3}\approx 0.225148227$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 1.5 by 1-x.

$$\left(1.5-1.5x\right)\left(1+x\right)+7x-3=0$$

Use the distributive property to multiply 1.5-1.5x by 1+x and combine like terms.

$$1.5-1.5x^{2}+7x-3=0$$

Subtract 3 from 1.5 to get -1.5.

$$-1.5-1.5x^{2}+7x=0$$

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.

$$-1.5x^{2}+7x-1.5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1.5 for a, 7 for b, and -1.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-1.5\right)\left(-1.5\right)}}{2\left(-1.5\right)}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-1.5\right)\left(-1.5\right)}}{2\left(-1.5\right)}$$

Multiply -4 times -1.5.

$$x=\frac{-7±\sqrt{49+6\left(-1.5\right)}}{2\left(-1.5\right)}$$

Multiply 6 times -1.5.

$$x=\frac{-7±\sqrt{49-9}}{2\left(-1.5\right)}$$

Add 49 to -9.

$$x=\frac{-7±\sqrt{40}}{2\left(-1.5\right)}$$

Take the square root of 40.

$$x=\frac{-7±2\sqrt{10}}{2\left(-1.5\right)}$$

Multiply 2 times -1.5.

$$x=\frac{-7±2\sqrt{10}}{-3}$$

Now solve the equation x=\frac{-7±2\sqrt{10}}{-3} when ± is plus. Add -7 to 2\sqrt{10}.

$$x=\frac{2\sqrt{10}-7}{-3}$$

Divide -7+2\sqrt{10} by -3.

$$x=\frac{7-2\sqrt{10}}{3}$$

Now solve the equation x=\frac{-7±2\sqrt{10}}{-3} when ± is minus. Subtract 2\sqrt{10} from -7.

$$x=\frac{-2\sqrt{10}-7}{-3}$$

Divide -7-2\sqrt{10} by -3.

$$x=\frac{2\sqrt{10}+7}{3}$$

The equation is now solved.

$$x=\frac{7-2\sqrt{10}}{3} x=\frac{2\sqrt{10}+7}{3}$$

Steps for Completing the Square

Use the distributive property to multiply 1.5 by 1-x.

$$\left(1.5-1.5x\right)\left(1+x\right)+7x-3=0$$

Use the distributive property to multiply 1.5-1.5x by 1+x and combine like terms.

$$1.5-1.5x^{2}+7x-3=0$$

Subtract 3 from 1.5 to get -1.5.

$$-1.5-1.5x^{2}+7x=0$$

Add 1.5 to both sides. Anything plus zero gives itself.

$$-1.5x^{2}+7x=1.5$$

Divide both sides of the equation by -1.5, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-1.5x^{2}+7x}{-1.5}=\frac{1.5}{-1.5}$$

Dividing by -1.5 undoes the multiplication by -1.5.

$$x^{2}+\frac{7}{-1.5}x=\frac{1.5}{-1.5}$$

Divide 7 by -1.5 by multiplying 7 by the reciprocal of -1.5.

$$x^{2}-\frac{14}{3}x=\frac{1.5}{-1.5}$$

Divide 1.5 by -1.5 by multiplying 1.5 by the reciprocal of -1.5.

$$x^{2}-\frac{14}{3}x=-1$$

Divide -\frac{14}{3}, the coefficient of the x term, by 2 to get -\frac{7}{3}. Then add the square of -\frac{7}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{14}{3}x+\left(-\frac{7}{3}\right)^{2}=-1+\left(-\frac{7}{3}\right)^{2}$$

Square -\frac{7}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{14}{3}x+\frac{49}{9}=-1+\frac{49}{9}$$

Add -1 to \frac{49}{9}.

$$x^{2}-\frac{14}{3}x+\frac{49}{9}=\frac{40}{9}$$

Factor x^{2}-\frac{14}{3}x+\frac{49}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{3}\right)^{2}=\frac{40}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{3}\right)^{2}}=\sqrt{\frac{40}{9}}$$

Simplify.

$$x-\frac{7}{3}=\frac{2\sqrt{10}}{3} x-\frac{7}{3}=-\frac{2\sqrt{10}}{3}$$

Add \frac{7}{3} to both sides of the equation.

$$x=\frac{2\sqrt{10}+7}{3} x=\frac{7-2\sqrt{10}}{3}$$