Solve (a-12)sqrt{(a+19+5)}=0

$$( a - 12 ) \sqrt { ( a + 19 + 5 ) } = 0$$

Solve for a

$$a=-24$$

$$a=12$$

Solution Steps

Add 19 and 5 to get 24.

$$\left(a-12\right)\sqrt{a+24}=0$$

Use the distributive property to multiply a-12 by \sqrt{a+24}.

$$a\sqrt{a+24}-12\sqrt{a+24}=0$$

Subtract -12\sqrt{a+24} from both sides of the equation.

$$a\sqrt{a+24}=12\sqrt{a+24}$$

Square both sides of the equation.

$$\left(a\sqrt{a+24}\right)^{2}=\left(12\sqrt{a+24}\right)^{2}$$

Expand \left(a\sqrt{a+24}\right)^{2}.

$$a^{2}\left(\sqrt{a+24}\right)^{2}=\left(12\sqrt{a+24}\right)^{2}$$

Calculate \sqrt{a+24} to the power of 2 and get a+24.

$$a^{2}\left(a+24\right)=\left(12\sqrt{a+24}\right)^{2}$$

Use the distributive property to multiply a^{2} by a+24.

$$a^{3}+24a^{2}=\left(12\sqrt{a+24}\right)^{2}$$

Expand \left(12\sqrt{a+24}\right)^{2}.

$$a^{3}+24a^{2}=12^{2}\left(\sqrt{a+24}\right)^{2}$$

Calculate 12 to the power of 2 and get 144.

$$a^{3}+24a^{2}=144\left(\sqrt{a+24}\right)^{2}$$

Calculate \sqrt{a+24} to the power of 2 and get a+24.

$$a^{3}+24a^{2}=144\left(a+24\right)$$

Use the distributive property to multiply 144 by a+24.

$$a^{3}+24a^{2}=144a+3456$$

Subtract 144a from both sides.

$$a^{3}+24a^{2}-144a=3456$$

Subtract 3456 from both sides.

$$a^{3}+24a^{2}-144a-3456=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -3456 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±3456,±1728,±1152,±864,±576,±432,±384,±288,±216,±192,±144,±128,±108,±96,±72,±64,±54,±48,±36,±32,±27,±24,±18,±16,±12,±9,±8,±6,±4,±3,±2,±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$a=12$$

By Factor theorem, a-k is a factor of the polynomial for each root k. Divide a^{3}+24a^{2}-144a-3456 by a-12 to get a^{2}+36a+288. Solve the equation where the result equals to 0.

$$a^{2}+36a+288=0$$

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, 36 for b, and 288 for c in the quadratic formula.

$$a=\frac{-36±\sqrt{36^{2}-4\times 1\times 288}}{2}$$

Do the calculations.

$$a=\frac{-36±12}{2}$$

Solve the equation a^{2}+36a+288=0 when ± is plus and when ± is minus.

$$a=-24 a=-12$$

List all found solutions.

$$a=12 a=-24 a=-12$$

Substitute 12 for a in the equation \left(a-12\right)\sqrt{a+19+5}=0.

$$\left(12-12\right)\sqrt{12+19+5}=0$$

Simplify. The value a=12 satisfies the equation.

$$0=0$$

Substitute -24 for a in the equation \left(a-12\right)\sqrt{a+19+5}=0.

$$\left(-24-12\right)\sqrt{-24+19+5}=0$$

Simplify. The value a=-24 satisfies the equation.

$$0=0$$

Substitute -12 for a in the equation \left(a-12\right)\sqrt{a+19+5}=0.

$$\left(-12-12\right)\sqrt{-12+19+5}=0$$

Simplify. The value a=-12 does not satisfy the equation.

$$-48\times 3^{\frac{1}{2}}=0$$

List all solutions of \sqrt{a+24}a=12\sqrt{a+24}.

$$a=12 a=-24$$