Solve (34/4-1.5x)(34/4+1.5x)=52

$$( \frac{ 34 }{ 4 } -1.5x)( \frac{ 34 }{ 4 } +1.5x)=52$$

Solve for x

$$x=3$$

$$x=-3$$

Steps by Finding Square Root

Reduce the fraction \frac{34}{4} to lowest terms by extracting and canceling out 2.

$$\left(\frac{17}{2}-1.5x\right)\left(\frac{34}{4}+1.5x\right)=52$$

Reduce the fraction \frac{34}{4} to lowest terms by extracting and canceling out 2.

$$\left(\frac{17}{2}-1.5x\right)\left(\frac{17}{2}+1.5x\right)=52$$

Consider \left(\frac{17}{2}-1.5x\right)\left(\frac{17}{2}+1.5x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square \frac{17}{2}.

$$\frac{289}{4}-\left(1.5x\right)^{2}=52$$

Expand \left(1.5x\right)^{2}.

$$\frac{289}{4}-1.5^{2}x^{2}=52$$

Calculate 1.5 to the power of 2 and get 2.25.

$$\frac{289}{4}-2.25x^{2}=52$$

Subtract \frac{289}{4} from both sides.

$$-2.25x^{2}=52-\frac{289}{4}$$

Subtract \frac{289}{4} from 52 to get -\frac{81}{4}.

$$-2.25x^{2}=-\frac{81}{4}$$

Divide both sides by -2.25.

$$x^{2}=\frac{-\frac{81}{4}}{-2.25}$$

Express \frac{-\frac{81}{4}}{-2.25} as a single fraction.

$$x^{2}=\frac{-81}{4\left(-2.25\right)}$$

Multiply 4 and -2.25 to get -9.

$$x^{2}=\frac{-81}{-9}$$

Divide -81 by -9 to get 9.

$$x^{2}=9$$

Take the square root of both sides of the equation.

$$x=3 x=-3$$

Steps Using the Quadratic Formula

Reduce the fraction \frac{34}{4} to lowest terms by extracting and canceling out 2.

$$\left(\frac{17}{2}-1.5x\right)\left(\frac{34}{4}+1.5x\right)=52$$

Reduce the fraction \frac{34}{4} to lowest terms by extracting and canceling out 2.

$$\left(\frac{17}{2}-1.5x\right)\left(\frac{17}{2}+1.5x\right)=52$$

Consider \left(\frac{17}{2}-1.5x\right)\left(\frac{17}{2}+1.5x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square \frac{17}{2}.

$$\frac{289}{4}-\left(1.5x\right)^{2}=52$$

Expand \left(1.5x\right)^{2}.

$$\frac{289}{4}-1.5^{2}x^{2}=52$$

Calculate 1.5 to the power of 2 and get 2.25.

$$\frac{289}{4}-2.25x^{2}=52$$

Subtract 52 from both sides.

$$\frac{289}{4}-2.25x^{2}-52=0$$

Subtract 52 from \frac{289}{4} to get \frac{81}{4}.

$$\frac{81}{4}-2.25x^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$-2.25x^{2}+\frac{81}{4}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2.25 for a, 0 for b, and \frac{81}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-2.25\right)\times \frac{81}{4}}}{2\left(-2.25\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-2.25\right)\times \frac{81}{4}}}{2\left(-2.25\right)}$$

Multiply -4 times -2.25.

$$x=\frac{0±\sqrt{9\times \frac{81}{4}}}{2\left(-2.25\right)}$$

Multiply 9 times \frac{81}{4}.

$$x=\frac{0±\sqrt{\frac{729}{4}}}{2\left(-2.25\right)}$$

Take the square root of \frac{729}{4}.

$$x=\frac{0±\frac{27}{2}}{2\left(-2.25\right)}$$

Multiply 2 times -2.25.

$$x=\frac{0±\frac{27}{2}}{-4.5}$$

Now solve the equation x=\frac{0±\frac{27}{2}}{-4.5} when ± is plus.

$$x=-3$$

Now solve the equation x=\frac{0±\frac{27}{2}}{-4.5} when ± is minus.

$$x=3$$

The equation is now solved.

$$x=-3 x=3$$