Solve 90x^2+x-1444=0

$$90 { x }^{ 2 } +x-1444=0$$

Solve for x

$$x = -\frac{361}{90} = -4\frac{1}{90} \approx -4.011111111$$

$$x=4$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 90x^{2}+ax+bx-1444. To find a and b, set up a system to be solved.

$$a+b=1 ab=90\left(-1444\right)=-129960$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -129960.

$$-1,129960 -2,64980 -3,43320 -4,32490 -5,25992 -6,21660 -8,16245 -9,14440 -10,12996 -12,10830 -15,8664 -18,7220 -19,6840 -20,6498 -24,5415 -30,4332 -36,3610 -38,3420 -40,3249 -45,2888 -57,2280 -60,2166 -72,1805 -76,1710 -90,1444 -95,1368 -114,1140 -120,1083 -152,855 -171,760 -180,722 -190,684 -228,570 -285,456 -342,380 -360,361$$

Calculate the sum for each pair.

$$-1+129960=129959 -2+64980=64978 -3+43320=43317 -4+32490=32486 -5+25992=25987 -6+21660=21654 -8+16245=16237 -9+14440=14431 -10+12996=12986 -12+10830=10818 -15+8664=8649 -18+7220=7202 -19+6840=6821 -20+6498=6478 -24+5415=5391 -30+4332=4302 -36+3610=3574 -38+3420=3382 -40+3249=3209 -45+2888=2843 -57+2280=2223 -60+2166=2106 -72+1805=1733 -76+1710=1634 -90+1444=1354 -95+1368=1273 -114+1140=1026 -120+1083=963 -152+855=703 -171+760=589 -180+722=542 -190+684=494 -228+570=342 -285+456=171 -342+380=38 -360+361=1$$

The solution is the pair that gives sum 1.

$$a=-360 b=361$$

Rewrite 90x^{2}+x-1444 as \left(90x^{2}-360x\right)+\left(361x-1444\right).

$$\left(90x^{2}-360x\right)+\left(361x-1444\right)$$

Factor out 90x in the first and 361 in the second group.

$$90x\left(x-4\right)+361\left(x-4\right)$$

Factor out common term x-4 by using distributive property.

$$\left(x-4\right)\left(90x+361\right)$$

To find equation solutions, solve x-4=0 and 90x+361=0.

$$x=4 x=-\frac{361}{90}$$

Steps Using the Quadratic Formula

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.

$$90x^{2}+x-1444=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 90 for a, 1 for b, and -1444 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\times 90\left(-1444\right)}}{2\times 90}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times 90\left(-1444\right)}}{2\times 90}$$

Multiply -4 times 90.

$$x=\frac{-1±\sqrt{1-360\left(-1444\right)}}{2\times 90}$$

Multiply -360 times -1444.

$$x=\frac{-1±\sqrt{1+519840}}{2\times 90}$$

Add 1 to 519840.

$$x=\frac{-1±\sqrt{519841}}{2\times 90}$$

Take the square root of 519841.

$$x=\frac{-1±721}{2\times 90}$$

Multiply 2 times 90.

$$x=\frac{-1±721}{180}$$

Now solve the equation x=\frac{-1±721}{180} when ± is plus. Add -1 to 721.

$$x=\frac{720}{180}$$

Divide 720 by 180.

$$x=4$$

Now solve the equation x=\frac{-1±721}{180} when ± is minus. Subtract 721 from -1.

$$x=-\frac{722}{180}$$

Reduce the fraction \frac{-722}{180} to lowest terms by extracting and canceling out 2.

$$x=-\frac{361}{90}$$

The equation is now solved.

$$x=4 x=-\frac{361}{90}$$

Steps for Completing the Square

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.

$$90x^{2}+x-1444=0$$

Add 1444 to both sides of the equation.

$$90x^{2}+x-1444-\left(-1444\right)=-\left(-1444\right)$$

Subtracting -1444 from itself leaves 0.

$$90x^{2}+x=-\left(-1444\right)$$

Subtract -1444 from 0.

$$90x^{2}+x=1444$$

Divide both sides by 90.

$$\frac{90x^{2}+x}{90}=\frac{1444}{90}$$

Dividing by 90 undoes the multiplication by 90.

$$x^{2}+\frac{1}{90}x=\frac{1444}{90}$$

Reduce the fraction \frac{1444}{90} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{1}{90}x=\frac{722}{45}$$

Divide \frac{1}{90}, the coefficient of the x term, by 2 to get \frac{1}{180}. Then add the square of \frac{1}{180} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{90}x+\left(\frac{1}{180}\right)^{2}=\frac{722}{45}+\left(\frac{1}{180}\right)^{2}$$

Square \frac{1}{180} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{90}x+\frac{1}{32400}=\frac{722}{45}+\frac{1}{32400}$$

Add \frac{722}{45} to \frac{1}{32400} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{90}x+\frac{1}{32400}=\frac{519841}{32400}$$

Factor x^{2}+\frac{1}{90}x+\frac{1}{32400}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{180}\right)^{2}=\frac{519841}{32400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{180}\right)^{2}}=\sqrt{\frac{519841}{32400}}$$

Simplify.

$$x+\frac{1}{180}=\frac{721}{180} x+\frac{1}{180}=-\frac{721}{180}$$

Subtract \frac{1}{180} from both sides of the equation.

$$x=4 x=-\frac{361}{90}$$