Solve (a^2-5+6)(a^2-64)=3465

$$( a ^ { 2 } - 5 + 6 ) ( a ^ { 2 } - 64 ) = 3465$$

Solve for a

$$a = -\frac{\sqrt{2 {(\sqrt{18085} + 63)}}}{2} \approx -9.936812413$$

$$a = \frac{\sqrt{2 {(\sqrt{18085} + 63)}}}{2} \approx 9.936812413$$

Steps Using the Quadratic Formula

Add -5 and 6 to get 1.

$$\left(a^{2}+1\right)\left(a^{2}-64\right)=3465$$

Use the distributive property to multiply a^{2}+1 by a^{2}-64 and combine like terms.

$$a^{4}-63a^{2}-64=3465$$

Subtract 3465 from both sides.

$$a^{4}-63a^{2}-64-3465=0$$

Subtract 3465 from -64 to get -3529.

$$a^{4}-63a^{2}-3529=0$$

Substitute t for a^{2}.

$$t^{2}-63t-3529=0$$

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, -63 for b, and -3529 for c in the quadratic formula.

$$t=\frac{-\left(-63\right)±\sqrt{\left(-63\right)^{2}-4\times 1\left(-3529\right)}}{2}$$

Do the calculations.

$$t=\frac{63±\sqrt{18085}}{2}$$

Solve the equation t=\frac{63±\sqrt{18085}}{2} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{18085}+63}{2} t=\frac{63-\sqrt{18085}}{2}$$

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

$$a=\frac{\sqrt{2\sqrt{18085}+126}}{2} a=-\frac{\sqrt{2\sqrt{18085}+126}}{2}$$