Solve (13^2-a^2)(a^2-3^2)=5000

$$( 13 ^ { 2 } - a ^ { 2 } ) ( a ^ { 2 } - 3 ^ { 2 } ) = 5000$$

Solve for a

$$a=-\sqrt{89-10\sqrt{14}}\approx -7.182160269$$

$$a=\sqrt{89-10\sqrt{14}}\approx 7.182160269$$

$$a=\sqrt{10\sqrt{14}+89}\approx 11.243512524$$

$$a=-\sqrt{10\sqrt{14}+89}\approx -11.243512524$$

Steps Using the Quadratic Formula

Calculate 13 to the power of 2 and get 169.

$$\left(169-a^{2}\right)\left(a^{2}-3^{2}\right)=5000$$

Calculate 3 to the power of 2 and get 9.

$$\left(169-a^{2}\right)\left(a^{2}-9\right)=5000$$

Use the distributive property to multiply 169-a^{2} by a^{2}-9 and combine like terms.

$$178a^{2}-1521-a^{4}=5000$$

Subtract 5000 from both sides.

$$178a^{2}-1521-a^{4}-5000=0$$

Subtract 5000 from -1521 to get -6521.

$$178a^{2}-6521-a^{4}=0$$

Substitute t for a^{2}.

$$-t^{2}+178t-6521=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute -1 for a, 178 for b, and -6521 for c in the quadratic formula.

$$t=\frac{-178±\sqrt{178^{2}-4\left(-1\right)\left(-6521\right)}}{-2}$$

Do the calculations.

$$t=\frac{-178±20\sqrt{14}}{-2}$$

Solve the equation t=\frac{-178±20\sqrt{14}}{-2} when ± is plus and when ± is minus.

$$t=89-10\sqrt{14} t=10\sqrt{14}+89$$

Since a=t^{2}, the solutions are obtained by evaluating a=±\sqrt{t} for each t.

$$a=\sqrt{89-10\sqrt{14}} a=-\sqrt{89-10\sqrt{14}} a=\sqrt{10\sqrt{14}+89} a=-\sqrt{10\sqrt{14}+89}$$