Solving Simple Quadratic Equations

To solve $x^2=25$, square root both sides, remembering that a positive number has two square roots, one positive and one negative.

$x=\pm \sqrt{25}=\pm 5$.

Both work: $5^2=25$ and $(-5{)}^2=25$. So the solutions are $x=5$ and $x=-5$.

$x^2=0$ means $x\times x=0$, and a product is zero only when one of the factors is zero, so the only solution is $x=0$.

Check it by squaring: $0^2=0$. Zero is a genuine solution: it is not the same as 'no solution'.

This is a repeated (double) root, so unlike $x^2=64$ there is one value, not a pair. And $x=1$ does not work, because $1^2=1$, not 0.

Squaring any real number gives an answer that is zero or positive: $4^2=16$ and $(-4{)}^2=16$, because two negatives multiply to give a positive.

So no real number can square to give $-16$, and $x^2=-16$ has no real solutions.

Watch the sign: the answers $x=4$ and $x=-4$ belong to the different equation $x^2=16$.

Add 64 to both sides to isolate the squared term: $x^2=64$.

Square-rooting gives two answers, because $8^2=64$ and $(-8{)}^2=64$. So $x=8$ and $x=-8$.

Two checks worth making: the $-64$ becomes $+64$ when it crosses the equals sign, and the square root must be taken, keeping both signs.

Add 81 to both sides to isolate the squared term: $x^2=81$.

Square-rooting gives two answers, because $9^2=81$ and $(-9{)}^2=81$. So $x=9$ and $x=-9$.

The final step really is a square root, not a halving: half of 81 is 40.5, and $40.5^2=1640.25$, nowhere near 81.

Add 125 to both sides: $5x^2=125$.

Divide both sides by 5 to leave the squared term on its own: $x^2=25$.

Square-rooting gives two answers, because $5^2=25$ and $(-5{)}^2=25$. So $x=-5$ and $x=5$.

Both final steps matter: divide by the 5 in front, then take the square root, keeping both signs.