Question

check your understanding
solve for x.

1. $(x - 3)^2=0$
2. $(3 - 5s)(-3 + 5s)=0$
3. $(d - 2)(d + 6)(d + 8)=0$
4. $x^{2}-144 = 0$
5. $g^{2}-13g + 40 = 0$
6. $7p^{2}=21p$
7. $x^{2}+3x=-2.$
8. $w^{3}-8w^{2}+16w = 0$
9. $x^{3}=25x$
10. $c^{3}-7c^{2}+12c = 0$
11. $3k^{3}-15k^{2}+18k = 0$
12. $6x^{3}-6x^{2}-12x = 0$

Explanation:

Step1: Solve equation (1)

If \((x - 3)^2=0\), then \(x-3 = 0\) (since if \(a^2 = 0\), then \(a=0\)). So \(x=3\).

Step2: Solve equation (2)

If \((3 - 5s)(-3 + 5s)=0\), then either \(3-5s = 0\) or \(-3 + 5s=0\).
For \(3-5s = 0\), we have \(5s=3\), so \(s=\frac{3}{5}\).
For \(-3 + 5s=0\), we have \(5s=3\), so \(s=\frac{3}{5}\).

Step3: Solve equation (3)

If \((d - 2)(d + 6)(d + 8)=0\), then \(d-2=0\) or \(d + 6=0\) or \(d+8=0\).
So \(d=2\) or \(d=-6\) or \(d=-8\).

Step4: Solve equation (4)

For \(x^{2}-144 = 0\), we can rewrite it as \(x^{2}=144\). Then \(x=\pm\sqrt{144}=\pm12\).

Step5: Solve equation (5)

For \(g^{2}-13g + 40=0\), we factor the quadratic: \((g - 5)(g - 8)=0\).
So \(g=5\) or \(g=8\).

Step6: Solve equation (6)

For \(7p^{2}=21p\), we move all terms to one - side: \(7p^{2}-21p=0\).
Factor out \(7p\): \(7p(p - 3)=0\).
So \(p = 0\) or \(p=3\).

Step7: Solve equation (7)

For \(x^{2}+3x=-2\), rewrite it in standard form \(x^{2}+3x + 2=0\).
Factor: \((x + 1)(x+2)=0\).
So \(x=-1\) or \(x=-2\).

Step8: Solve equation (8)

For \(w^{3}-8w^{2}+16w=0\), factor out \(w\): \(w(w^{2}-8w + 16)=0\).
Factor the quadratic: \(w(w - 4)^{2}=0\).
So \(w=0\) or \(w = 4\).

Step9: Solve equation (9)

For \(x^{3}=25x\), move all terms to one - side: \(x^{3}-25x=0\).
Factor out \(x\): \(x(x^{2}-25)=0\).
Factor the difference of squares: \(x(x - 5)(x + 5)=0\).
So \(x=0\) or \(x=5\) or \(x=-5\).

Step10: Solve equation (10)

For \(c^{3}-7c^{2}+12c=0\), factor out \(c\): \(c(c^{2}-7c + 12)=0\).
Factor the quadratic: \(c(c - 3)(c - 4)=0\).
So \(c=0\) or \(c=3\) or \(c=4\).

Step11: Solve equation (11)

For \(3k^{3}-15k^{2}+18k=0\), factor out \(3k\): \(3k(k^{2}-5k + 6)=0\).
Factor the quadratic: \(3k(k - 2)(k - 3)=0\).
So \(k=0\) or \(k=2\) or \(k=3\).

Step12: Solve equation (12)

For \(6x^{3}-6x^{2}-12x=0\), factor out \(6x\): \(6x(x^{2}-x - 2)=0\).
Factor the quadratic: \(6x(x - 2)(x+1)=0\).
So \(x=0\) or \(x=2\) or \(x=-1\).

1. \(x = 3\)
2. \(s=\frac{3}{5}\)
3. \(d=2\), \(d=-6\), \(d=-8\)
4. \(x = 12\), \(x=-12\)
5. \(g=5\), \(g=8\)
6. \(p = 0\), \(p=3\)
7. \(x=-1\), \(x=-2\)
8. \(w=0\), \(w = 4\)
9. \(x=0\), \(x=5\), \(x=-5\)
10. \(c=0\), \(c=3\), \(c=4\)
11. \(k=0\), \(k=2\), \(k=3\)
12. \(x=0\), \(x=2\), \(x=-1\)

Answer:

Step1: Solve equation (1)

If \((x - 3)^2=0\), then \(x-3 = 0\) (since if \(a^2 = 0\), then \(a=0\)). So \(x=3\).

Step2: Solve equation (2)

If \((3 - 5s)(-3 + 5s)=0\), then either \(3-5s = 0\) or \(-3 + 5s=0\).
For \(3-5s = 0\), we have \(5s=3\), so \(s=\frac{3}{5}\).
For \(-3 + 5s=0\), we have \(5s=3\), so \(s=\frac{3}{5}\).

Step3: Solve equation (3)

If \((d - 2)(d + 6)(d + 8)=0\), then \(d-2=0\) or \(d + 6=0\) or \(d+8=0\).
So \(d=2\) or \(d=-6\) or \(d=-8\).

Step4: Solve equation (4)

For \(x^{2}-144 = 0\), we can rewrite it as \(x^{2}=144\). Then \(x=\pm\sqrt{144}=\pm12\).

Step5: Solve equation (5)

For \(g^{2}-13g + 40=0\), we factor the quadratic: \((g - 5)(g - 8)=0\).
So \(g=5\) or \(g=8\).

Step6: Solve equation (6)

For \(7p^{2}=21p\), we move all terms to one - side: \(7p^{2}-21p=0\).
Factor out \(7p\): \(7p(p - 3)=0\).
So \(p = 0\) or \(p=3\).

Step7: Solve equation (7)

For \(x^{2}+3x=-2\), rewrite it in standard form \(x^{2}+3x + 2=0\).
Factor: \((x + 1)(x+2)=0\).
So \(x=-1\) or \(x=-2\).

Step8: Solve equation (8)

For \(w^{3}-8w^{2}+16w=0\), factor out \(w\): \(w(w^{2}-8w + 16)=0\).
Factor the quadratic: \(w(w - 4)^{2}=0\).
So \(w=0\) or \(w = 4\).

Step9: Solve equation (9)

For \(x^{3}=25x\), move all terms to one - side: \(x^{3}-25x=0\).
Factor out \(x\): \(x(x^{2}-25)=0\).
Factor the difference of squares: \(x(x - 5)(x + 5)=0\).
So \(x=0\) or \(x=5\) or \(x=-5\).

Step10: Solve equation (10)

For \(c^{3}-7c^{2}+12c=0\), factor out \(c\): \(c(c^{2}-7c + 12)=0\).
Factor the quadratic: \(c(c - 3)(c - 4)=0\).
So \(c=0\) or \(c=3\) or \(c=4\).

Step11: Solve equation (11)

For \(3k^{3}-15k^{2}+18k=0\), factor out \(3k\): \(3k(k^{2}-5k + 6)=0\).
Factor the quadratic: \(3k(k - 2)(k - 3)=0\).
So \(k=0\) or \(k=2\) or \(k=3\).

Step12: Solve equation (12)

For \(6x^{3}-6x^{2}-12x=0\), factor out \(6x\): \(6x(x^{2}-x - 2)=0\).
Factor the quadratic: \(6x(x - 2)(x+1)=0\).
So \(x=0\) or \(x=2\) or \(x=-1\).

1. \(x = 3\)
2. \(s=\frac{3}{5}\)
3. \(d=2\), \(d=-6\), \(d=-8\)
4. \(x = 12\), \(x=-12\)
5. \(g=5\), \(g=8\)
6. \(p = 0\), \(p=3\)
7. \(x=-1\), \(x=-2\)
8. \(w=0\), \(w = 4\)
9. \(x=0\), \(x=5\), \(x=-5\)
10. \(c=0\), \(c=3\), \(c=4\)
11. \(k=0\), \(k=2\), \(k=3\)
12. \(x=0\), \(x=2\), \(x=-1\)