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Section 4.4 Solving Quadratic Equations by Factoring
If a quadratic equation is factored, then it is easy to solve.
How to factor quadratics in the form \(ax^2 + bx + c\text{.}\)
Combining these give us a powerful strategy for solving quadratic equations. First however, we need to make sure the equation is in the correct form,
\begin{gather*}
ax^2 + bx + c = 0
\end{gather*}
This is called
standard form , as it is the conventional, agreed-upon way to write a quadratic equation. This is because this form makes solving the equation consistent each time. To do this, move all terms to one side of the equation, and arrange the terms with the
\(x^2\) first, the
\(x\) term next, and the constant last.
Subsection 4.4.1 Summary
(If necessary) Write in standard form, by moving all terms to one side so that the equation equals zero. E.g. \(x^2 + 5x = 14 \rightarrow x^2 + 5x - 14 = 0\)
(If necessary) Look for common factors, if all terms share a common factor, divide it out first.
Factor the trinomial.
Find the value of \(x\) that makes each factor 0.
If \(x\) is a factor of the polynomial, then \(x = 0\) is a solution to the equation.
Always look for common factors, because it makes the problem more simple.
Exercise Group 4.4.1 . Solving Quadratics by Factoring.
(a)
\(x^2 + 3x + 2 = 0\text{.}\)
(b)
\(x^2 + 5x + 6 = 0\text{.}\)
(c)
\(x^2 - 3x + 2 = 0\text{.}\)
(d)
\(x^2 - x - 12 = 0\text{.}\)
(e)
\(2x^2 + 18x + 36 = 0\text{.}\)
(f) (g)
\(x^2 + 7x = -6\text{.}\)
(h)
\(-x^2 - 11x - 28 = 0\text{.}\)
(i)
\(x^2 = 14x - 49\text{.}\)
(j)
\(x^2 + 14 = -9x\text{.}\)
(k)
\(4x^2 + 24 = 20x\text{.}\)
(l)
\(x^2 + 11x = -30\text{.}\)
(m)
\(x^2 + 4x = 12\text{.}\)
(n)
\(x^2 = 11x - 24\text{.}\)
(o)
\(2x^2 + 2x = 24\text{.}\)
(p)
\(x^2 + 6x + 8 = 0\text{.}\)
(q)
\(-x^2 = 7x - 60\text{.}\)
(r)
\(x^2 + 4x + 4 = 0\text{.}\)
(s)
\(x^2 = 3x + 18\text{.}\)
(t)
\(x^2 = 5x + 14\text{.}\)
(u)
\(x^2 + 60 = 17x\text{.}\)
(v)
\(x^2 = 9x + 22\text{.}\)
(w)
\(x^2 = 7x - 12\text{.}\)
(x)
\(x^2 + 25 = 10x\text{.}\)
(y)
\(x^2 + 11x + 24 = 0\text{.}\)
(z)
\(x^2 - 6x + 9 = 0\text{.}\)
(aa)
\(140 + 6x = 2x^2\text{.}\)
(ab)
\(3x^2 + 6x = 24\text{.}\)
(ac)
\(4x^2 - 28x = 0\text{.}\)
(ad)
\(5x^2 + 20 = 20x\text{.}\)
(ae)
\(3x^2 = 3x + 18\text{.}\)
(af)
\(2x^2 + 4x - 16 = 0\text{.}\)
Subsection 4.4.2 Equations Requiring the Cross Method
Exercise Group 4.4.2 . Solving Quadratics with Leading Coefficient.
(a)
\(2x^2 - 5x + 2 = 0\text{.}\)
Answer .
\(x = 2, \frac{1}{2}\text{.}\)
(b)
\(3x^2 + 8x - 3 = 0\text{.}\)
Answer .
\(x = \frac{1}{3}, -3\text{.}\)
(c)
\(3x^2 + 17x + 20 = 0\text{.}\)
Answer .
\(x = -\frac{5}{3}, -4\text{.}\)
(d)
\(2x^2 + 5x = 3\text{.}\)
Answer .
\(x = \frac{1}{2}, -3\text{.}\)
(e)
\(2x^2 + 5 = 11x\text{.}\)
Answer .
\(x = 5, \frac{1}{2}\text{.}\)
(f)
\(2x^2 + 7x + 5 = 0\text{.}\)
Answer .
\(x = -1, -\frac{5}{2}\text{.}\)
(g)
\(10x^2 + 5 = 27x\text{.}\)
Answer .
\(x = \frac{5}{2}, \frac{1}{5}\text{.}\)
(h)
\(25x^2 + 10x = 8\text{.}\)
Answer .
\(x = \frac{2}{5}, -\frac{4}{5}\text{.}\)
(i)
\(2x^2 + 11x + 5 = 0\text{.}\)
Answer .
\(x = -\frac{1}{2}, -5\text{.}\)
(j)
\(5x^2 - 21x + 4 = 0\text{.}\)
Answer .
\(x = 4, \frac{1}{5}\text{.}\)
(k)
\(3x^2 = 11x + 4\text{.}\)
Answer .
\(x = 4, -\frac{1}{3}\text{.}\)
(l)
\(3x^2 + 10x = 8\text{.}\)
Answer .
\(x = \frac{2}{3}, -4\text{.}\)
(m)
\(2x^2 = 13x + 7\text{.}\)
Answer .
\(x = 7, -\frac{1}{2}\text{.}\)
(n)
\(7x^2 = 11x + 6\text{.}\)
Answer .
\(x = 2, -\frac{3}{7}\text{.}\)
(o)
\(3x^2 + 10 = 17x\text{.}\)
Answer .
\(x = 5, \frac{2}{3}\text{.}\)
(p)
\(3x^2 + x = 10\text{.}\)
Answer .
\(x = \frac{5}{3}, -2\text{.}\)
(q)
\(2x^2 = 13x + 15\text{.}\)
Answer .
\(x = \frac{15}{2}, -1\text{.}\)
(r)
\(2x^2 = 7x + 15\text{.}\)
Answer .
\(x = 5, -\frac{3}{2}\text{.}\)
(s)
\(3x^2 - 7x + 2 = 0\text{.}\)
Answer .
\(x = \frac{1}{3}, 2\text{.}\)
(t)
\(6x^2 + 11x - 35=0\text{.}\)
Answer .
\(x = \frac{5}{3}, -\frac{7}{2}\text{.}\)
(u)
\(2x^2 + 17x = 9\text{.}\)
Answer .
\(x = \frac{1}{2}, -9\text{.}\)
(v)
\(6x^2 + 11x + 3 = 0\text{.}\)
Answer .
\(x = -\frac{1}{3}, -\frac{3}{2}\text{.}\)
(w)
\(6x^2 = 17x + 3\text{.}\)
Answer .
\(x = 3, -\frac{1}{6}\text{.}\)
(x) Answer .
\(x = \frac{1}{2}, -\frac{2}{3}\text{.}\)
(y)
\(10x^2 + 21x = 10\text{.}\)
Answer .
\(x = \frac{2}{5}, -\frac{5}{2}\text{.}\)
(z)
\(12x^2 + 13x + 3 = 0\text{.}\)
Answer .
\(x = -\frac{1}{3}, -\frac{3}{4}\text{.}\)
(aa)
\(6x^2 = 17x + 14\text{.}\)
Answer .
\(x = \frac{7}{2}, -\frac{2}{3}\text{.}\)
(ab)
\(14x^2 + 15x + 4 = 0\text{.}\)
Answer .
\(x = -\frac{1}{2}, -\frac{4}{7}\text{.}\)
(ac)
\(6x^2 - 13x + 6 = 0\text{.}\)
Answer .
\(x = \frac{3}{2}, \frac{2}{3}\text{.}\)
(ad)
\(12x^2 + 13x = 4\text{.}\)
Answer .
\(x = \frac{1}{4}, -\frac{4}{3}\text{.}\)
(ae)
\(15x^2 + 6 = 23x\text{.}\)
Answer .
\(x = \frac{6}{5}, \frac{1}{3}\text{.}\)
(af)
\(8x^2 = 10x + 3\text{.}\)
Answer .
\(x = \frac{3}{2}, -\frac{1}{4}\text{.}\)
(ag)
\(2x^2 + 5x = 12\text{.}\)
Answer .
\(x = \frac{3}{2}, -4\text{.}\)
(ah)
\(3x^2 + 2x = 16\text{.}\)
Answer .
\(x = 2, -\frac{8}{3}\text{.}\)
Exercise Group 4.4.3 . Advanced Quadratic Equations.
Advanced examples . Solve each equation.
(a)
\(x(3 + x) + 3 = 31\text{.}\)
(b)
\(3x(x + 2) = 9\text{.}\)
(c)
\(2x(x - 1) - 3(x + 2) = -3\text{.}\)
Answer .
\(x = 3, -\frac{1}{2}\text{.}\)
(d)
\(56x^2 = 23x + 63\text{.}\)
Answer .
\(x = \frac{9}{7}, -\frac{7}{8}\text{.}\)
(e)
\(6 - 35x - 6x^2 = 0\text{.}\)
Answer .
\(x = \frac{1}{6}, -6\text{.}\)
(f)
\(x + 32 = x(x - 3)\text{.}\)
(g)
\(x^2 - 3x + 26 = (2x - 3)(x + 2)\text{.}\)
(h)
\((3x - 2)(x + 2) = x^2 - 11x - 22\text{.}\)
Answer .
\(x = -\frac{3}{2}, -6\text{.}\)
(i) (j) Answer .
\(x = 6, \frac{2}{3}\text{.}\)
(k)
\((x-12)(x+1)=-40\text{.}\)
(l)
\((x+3)(x+5)=5x+25\text{.}\)
(m)
\((4x-5)(x-5) = -45x\text{.}\)
Answer .
\(x = -\frac{5}{2}\text{.}\)
(n)
\((x+1)(x-6) = -10\text{.}\)
(o)
\(\frac{x^2}{18} + \frac{x}{6} = 1\text{.}\)
(p)
\(3x^2 + 13x + 4 = 0\text{.}\)
Answer .
\(x = -4, -\frac{1}{3}\text{.}\)
(q)
\(5x^2 - 6 = 13x\text{.}\)
Answer .
\(x = -\frac{2}{5}, 3\text{.}\)
(r)
\(2x^2 = 3x + 5\text{.}\)
Answer .
\(x = -1, \frac{5}{2}\text{.}\)
(s)
\(3x^2 = 8 - 2x\text{.}\)
Answer .
\(x = -2, \frac{4}{3}\text{.}\)
(t)
\(2x^2 = 18 - 9x\text{.}\)
Answer .
\(x = -6, \frac{3}{2}\text{.}\)
(u)
\(-2x^2 - 5x + 12 = 0\text{.}\)
Answer .
\(x = -4, \frac{3}{2}\text{.}\)
(v)
\(12x^2 - 22x + 6 = 0\text{.}\)
Answer .
\(x = \frac{1}{3}, \frac{3}{2}\text{.}\)
(w)
\(9x^2 + 6x - 48 = 0\text{.}\)
Answer .
\(x = -\frac{8}{3}, 2\text{.}\)
(x)
\(28x^2 - 8 = 2x\text{.}\)
Answer .
\(x = -\frac{1}{2}, \frac{4}{7}\text{.}\)
(y)
\(36x^2 + 39x = 12\text{.}\)
Answer .
\(x = -\frac{4}{3}, \frac{1}{4}\text{.}\)
