In Exercises solvequadfirst - solvequadlast, find all real solutions. Check your answers, as directed by your instructor.
\(3\left (x - \frac {1}{2}\right )^2 = \frac {5}{12}\)

\(x = \frac {3 \pm \sqrt {5}}{6}\)
\(4 - (5t+3)^2 = 3\)

\(t = \answer {-\frac {4}{5}}\) (smaller solution)

\(t = \answer {-\frac {2}{5}}\) (larger solution)

\(3(y^2-3)^2-2 = 10\)

\(y = \pm 1\), \(\pm \sqrt {5}\)
\(x^2 + x - 1 = 0\)

\(x = \frac {-1 \pm \sqrt {5}}{2}\)
\(3w^2 = 2-w\)

\(w = \answer {-1}\) (smaller solution)

\(w = \answer {\frac {2}{3}}\) (larger solution)

\(y(y+4) = 1\)

\(y = -2 \pm \sqrt {5}\)
\(\frac {z}{2} = 4z^2-1\)

\(z = \frac {1 \pm \sqrt {\answer {65}}}{16}\)

\(0.1v^2 + 0.2v = 0.3\)

\(v = \answer {-3}\) (smaller solution)

\(v= \answer {1}\) (larger solution)

\(x^2 = x - 1\)

No real solutions.
\(3-t = 2(t+1)^2\)

\(t = \frac {-5 \pm \sqrt {33}}{4}\)
\((x-3)^2 = x^2+9\)

\(x = \answer {0}\)

\((3y-1)(2y+1) = 5y\)

\(y = \frac {2 \pm \sqrt {10}}{6}\)
\(w^4 + 3w^2 - 1 = 0\)

\(w = \pm \sqrt {\frac {\sqrt {13} - 3}{2}}\)
\(2x^4 +x^2 = 3\)

\(x = \answer {-1}\) (smaller solution)

\(x = \answer {1}\) (larger solution)

\((2-y)^4 = 3(2-y)^2 + 1\)

\(y = \frac {4 \pm \sqrt {6 + 2 \sqrt {13}}}{2}\)
\(3x^4 + 6x^2 = 15x^3\)

\(x = 0, \frac {5 \pm \sqrt {17}}{2}\)
\(6p + 2 = p^2 + 3p^3\)

\(p = -\frac {1}{3}, \pm \sqrt {2}\)
\(10v = 7v^3 - v^5\)

\(v = 0, \pm \sqrt {2}, \pm \sqrt {5}\)
\(y^2 - \sqrt {8} y = \sqrt {18} y - 1\)

\(y = \frac {5\sqrt {2} \pm \sqrt {46}}{2}\)
\(x^2 \sqrt {3} = x \sqrt {6} + \sqrt {12}\)

\(x = \frac {\sqrt {2} \pm \sqrt {10}}{2}\)
\(\frac {v^2}{3} = \frac {v \sqrt {3}}{2} + 1\)

\(v = \answer {-\frac {\sqrt {3}}{2}}\) (smaller solution)

\(v = \answer {2\sqrt {3}}\) (larger solution)

In Exercises solvequadcalcfirst - solvequadcalclast, find all real solutions and use a calculator to approximate your answers, rounded to two decimal places.

\(5.54^2 + b^2 = 36\)

\(b = \pm \frac {\sqrt {13271}}{50} \approx \pm 2.30\)
\(\pi r^2 = 37\)

\(r = \pm \sqrt {\frac {37}{\pi }} \approx \pm 3.43\)
\(54 = 8r\sqrt {2} + \pi r^2\)

\(r = \frac {-4\sqrt {2} \pm \sqrt {54\pi + 32}}{\pi }\), \(r \approx -6.32, 2.72\)
\(-4.9t^2 + 100t = 410\)

\(t = \frac {500 \pm 10\sqrt {491}}{49}\), \(t \approx 5.68, 14.73\)
\(x^2 = 1.65(3-x)^2\)

\(x = \frac {99 \pm 6 \sqrt {165}}{13}\), \(x \approx 1.69, 13.54\)
\((0.5+2A)^2 = 0.7(0.1-A)^2\)

\(A = \frac {-107 \pm 7 \sqrt {70}}{330}\), \(A \approx -0.50, -0.15\)
In Exercises absquadfirst - absquadlast, use Theorem absvalequality along with the techniques in this section to find all real solutions to the following.
\(|x^2 - 3x| = 2\)

\(x = 1, 2, \frac {3 \pm \sqrt {17}}{2}\)
\(|2x-x^2| = |2x-1|\)

\(x = \pm 1, 2 \pm \sqrt {3}\)
\(|x^2 -x + 3| = |4-x^2|\)

\(x = -\frac {1}{2}, 1, 7\)
Prove that for every nonzero number \(p\), \(x^2 + xp + p^2 = 0\) has no real solutions.

The discriminant is: \(D = p^2 - 4p^2 = -3p^2 < 0\). Since \(D < 0\), there are no real solutions.
Solve for \(t\): \(-\frac {1}{2} g t^2 + vt + h = 0\). Assume \(g > 0\), \(v \geq 0\) and \(h \geq 0\).

\(t = \frac {v \pm \sqrt {v^2 + 2gh}}{g}\)