Solve (4x+3)(7x+5)=2

$$( 4 x + 3 ) ( 7 x + 5 ) = 2$$

Solve for x

$$x=-\frac{13}{28}\approx -0.464285714$$

$$x=-1$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4x+3 by 7x+5 and combine like terms.

$$28x^{2}+41x+15=2$$

Subtract 2 from both sides.

$$28x^{2}+41x+15-2=0$$

Subtract 2 from 15 to get 13.

$$28x^{2}+41x+13=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 28 for a, 41 for b, and 13 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-41±\sqrt{41^{2}-4\times 28\times 13}}{2\times 28}$$

Square 41.

$$x=\frac{-41±\sqrt{1681-4\times 28\times 13}}{2\times 28}$$

Multiply -4 times 28.

$$x=\frac{-41±\sqrt{1681-112\times 13}}{2\times 28}$$

Multiply -112 times 13.

$$x=\frac{-41±\sqrt{1681-1456}}{2\times 28}$$

Add 1681 to -1456.

$$x=\frac{-41±\sqrt{225}}{2\times 28}$$

Take the square root of 225.

$$x=\frac{-41±15}{2\times 28}$$

Multiply 2 times 28.

$$x=\frac{-41±15}{56}$$

Now solve the equation x=\frac{-41±15}{56} when ± is plus. Add -41 to 15.

$$x=-\frac{26}{56}$$

Reduce the fraction \frac{-26}{56} to lowest terms by extracting and canceling out 2.

$$x=-\frac{13}{28}$$

Now solve the equation x=\frac{-41±15}{56} when ± is minus. Subtract 15 from -41.

$$x=-\frac{56}{56}$$

Divide -56 by 56.

$$x=-1$$

The equation is now solved.

$$x=-\frac{13}{28} x=-1$$

Steps for Completing the Square

Use the distributive property to multiply 4x+3 by 7x+5 and combine like terms.

$$28x^{2}+41x+15=2$$

Subtract 15 from both sides.

$$28x^{2}+41x=2-15$$

Subtract 15 from 2 to get -13.

$$28x^{2}+41x=-13$$

Divide both sides by 28.

$$\frac{28x^{2}+41x}{28}=-\frac{13}{28}$$

Dividing by 28 undoes the multiplication by 28.

$$x^{2}+\frac{41}{28}x=-\frac{13}{28}$$

Divide \frac{41}{28}, the coefficient of the x term, by 2 to get \frac{41}{56}. Then add the square of \frac{41}{56} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{41}{28}x+\left(\frac{41}{56}\right)^{2}=-\frac{13}{28}+\left(\frac{41}{56}\right)^{2}$$

Square \frac{41}{56} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{41}{28}x+\frac{1681}{3136}=-\frac{13}{28}+\frac{1681}{3136}$$

Add -\frac{13}{28} to \frac{1681}{3136} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{41}{28}x+\frac{1681}{3136}=\frac{225}{3136}$$

Factor x^{2}+\frac{41}{28}x+\frac{1681}{3136}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{41}{56}\right)^{2}=\frac{225}{3136}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{41}{56}\right)^{2}}=\sqrt{\frac{225}{3136}}$$

Simplify.

$$x+\frac{41}{56}=\frac{15}{56} x+\frac{41}{56}=-\frac{15}{56}$$

Subtract \frac{41}{56} from both sides of the equation.

$$x=-\frac{13}{28} x=-1$$