Solve {l}{4ac-b^2/4a=x}{a=-3/2}{b=1/2}{c=9/2}

$$\left\{ \begin{array} { l } { \frac { 4 a c - b ^ { 2 } } { 4 a } = x } \\ { a = - \frac { 3 } { 2 } } \\ { b = \frac { 1 } { 2 } } \\ { c = \frac { 9 } { 2 } } \end{array} \right.$$

Solve for a, c, b, x

$$x = \frac{109}{24} = 4\frac{13}{24} \approx 4.541666667$$

$$a = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

$$c = \frac{9}{2} = 4\frac{1}{2} = 4.5$$

$$b=\frac{1}{2}=0.5$$

Solution Steps

Consider the first equation. Insert the known values of variables into the equation.

$$\frac{4\left(-\frac{3}{2}\right)\times \frac{9}{2}-\left(\frac{1}{2}\right)^{2}}{4\left(-\frac{3}{2}\right)}=x$$

Multiply 4 and -\frac{3}{2} to get -6.

$$\frac{-6\times \frac{9}{2}-\left(\frac{1}{2}\right)^{2}}{4\left(-\frac{3}{2}\right)}=x$$

Multiply -6 and \frac{9}{2} to get -27.

$$\frac{-27-\left(\frac{1}{2}\right)^{2}}{4\left(-\frac{3}{2}\right)}=x$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\frac{-27-\frac{1}{4}}{4\left(-\frac{3}{2}\right)}=x$$

Subtract \frac{1}{4} from -27 to get -\frac{109}{4}.

$$\frac{-\frac{109}{4}}{4\left(-\frac{3}{2}\right)}=x$$

Multiply 4 and -\frac{3}{2} to get -6.

$$\frac{-\frac{109}{4}}{-6}=x$$

Express \frac{-\frac{109}{4}}{-6} as a single fraction.

$$\frac{-109}{4\left(-6\right)}=x$$

Multiply 4 and -6 to get -24.

$$\frac{-109}{-24}=x$$

Fraction \frac{-109}{-24} can be simplified to \frac{109}{24} by removing the negative sign from both the numerator and the denominator.

$$\frac{109}{24}=x$$

Swap sides so that all variable terms are on the left hand side.

$$x=\frac{109}{24}$$

The system is now solved.

$$a=-\frac{3}{2} c=\frac{9}{2} b=\frac{1}{2} x=\frac{109}{24}$$