Solve x^2+x^2*27-x*9=19

$${ x }^{ 2 } + { x }^{ 2 } \times 27-x \times 9=19$$

Solve for x

$$x=-\frac{19}{28}\approx -0.678571429$$

$$x=1$$

Steps Using Factoring By Grouping

Combine x^{2} and x^{2}\times 27 to get 28x^{2}.

$$28x^{2}-x\times 9=19$$

Subtract 19 from both sides.

$$28x^{2}-x\times 9-19=0$$

Multiply -1 and 9 to get -9.

$$28x^{2}-9x-19=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 28x^{2}+ax+bx-19. To find a and b, set up a system to be solved.

$$a+b=-9 ab=28\left(-19\right)=-532$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -532.

$$1,-532 2,-266 4,-133 7,-76 14,-38 19,-28$$

Calculate the sum for each pair.

$$1-532=-531 2-266=-264 4-133=-129 7-76=-69 14-38=-24 19-28=-9$$

The solution is the pair that gives sum -9.

$$a=-28 b=19$$

Rewrite 28x^{2}-9x-19 as \left(28x^{2}-28x\right)+\left(19x-19\right).

$$\left(28x^{2}-28x\right)+\left(19x-19\right)$$

Factor out 28x in the first and 19 in the second group.

$$28x\left(x-1\right)+19\left(x-1\right)$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(28x+19\right)$$

To find equation solutions, solve x-1=0 and 28x+19=0.

$$x=1 x=-\frac{19}{28}$$

Steps Using the Quadratic Formula

Combine x^{2} and x^{2}\times 27 to get 28x^{2}.

$$28x^{2}-x\times 9=19$$

Subtract 19 from both sides.

$$28x^{2}-x\times 9-19=0$$

Multiply -1 and 9 to get -9.

$$28x^{2}-9x-19=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 28 for a, -9 for b, and -19 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 28\left(-19\right)}}{2\times 28}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 28\left(-19\right)}}{2\times 28}$$

Multiply -4 times 28.

$$x=\frac{-\left(-9\right)±\sqrt{81-112\left(-19\right)}}{2\times 28}$$

Multiply -112 times -19.

$$x=\frac{-\left(-9\right)±\sqrt{81+2128}}{2\times 28}$$

Add 81 to 2128.

$$x=\frac{-\left(-9\right)±\sqrt{2209}}{2\times 28}$$

Take the square root of 2209.

$$x=\frac{-\left(-9\right)±47}{2\times 28}$$

The opposite of -9 is 9.

$$x=\frac{9±47}{2\times 28}$$

Multiply 2 times 28.

$$x=\frac{9±47}{56}$$

Now solve the equation x=\frac{9±47}{56} when ± is plus. Add 9 to 47.

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

Divide 56 by 56.

$$x=1$$

Now solve the equation x=\frac{9±47}{56} when ± is minus. Subtract 47 from 9.

$$x=-\frac{38}{56}$$

Reduce the fraction \frac{-38}{56} to lowest terms by extracting and canceling out 2.

$$x=-\frac{19}{28}$$

The equation is now solved.

$$x=1 x=-\frac{19}{28}$$

Steps for Completing the Square

Combine x^{2} and x^{2}\times 27 to get 28x^{2}.

$$28x^{2}-x\times 9=19$$

Multiply -1 and 9 to get -9.

$$28x^{2}-9x=19$$

Divide both sides by 28.

$$\frac{28x^{2}-9x}{28}=\frac{19}{28}$$

Dividing by 28 undoes the multiplication by 28.

$$x^{2}-\frac{9}{28}x=\frac{19}{28}$$

Divide -\frac{9}{28}, the coefficient of the x term, by 2 to get -\frac{9}{56}. Then add the square of -\frac{9}{56} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{28}x+\left(-\frac{9}{56}\right)^{2}=\frac{19}{28}+\left(-\frac{9}{56}\right)^{2}$$

Square -\frac{9}{56} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{28}x+\frac{81}{3136}=\frac{19}{28}+\frac{81}{3136}$$

Add \frac{19}{28} to \frac{81}{3136} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{9}{28}x+\frac{81}{3136}=\frac{2209}{3136}$$

Factor x^{2}-\frac{9}{28}x+\frac{81}{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}.

$$\left(x-\frac{9}{56}\right)^{2}=\frac{2209}{3136}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{56}\right)^{2}}=\sqrt{\frac{2209}{3136}}$$

Simplify.

$$x-\frac{9}{56}=\frac{47}{56} x-\frac{9}{56}=-\frac{47}{56}$$

Add \frac{9}{56} to both sides of the equation.

$$x=1 x=-\frac{19}{28}$$