Algebra 2 by Richard Wright
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Objectives:
SDA NAD Content Standards (2018): AII.4.1, AII.4.2, AII.5.1, AII.6.3
The last several lessons have gone over several methods of how to solve quadratic equations. The problem is then which method to use.
Usually in applications of quadratic equations, the method to solve it is not given. The person solving has to choose. Because the quadratic formula is derived from completing the square, it is usually faster, and thus, preferred.
To decide what method to use to solve quadratic equations, it is usually fastest to try factoring or square roots first.
To most efficiently solve a quadratic equation,
Solve x2 − 18x + 81 = 0.
Solution
Both x2 and x appear. The equation already equals zero.
Try solving by factoring.
x2 – 18x + 81 = 0
(x – 9)(x – 9) = 0
x – 9 = 0
x = 9
Solve x2 = 11x – 24
Answer
(Factoring) 3, 8
Solve 3x2 = x + 14
Solution
Both x2 and x appear. Make the equation equal to zero.
3x2 = x + 14
3x2 – x – 14 = 0
Both x2 and x appear. The equation already equals zero.
Try solving by factoring.
(3x – 7)(x + 2) = 0
3x − 7 = 0 or x + 2 = 0
3x = 7 or x = −2
x = \(\frac{\mathbf{7}}{\mathbf{3}}\) or x = −2
Solve x2 − 6x = 0
Answer
(Factoring) 0, 6
Solve 3x2 = 147.
Solution
Only x2 appears, so solve by square roots.
3x2 = 147
x2 = 49
x = ±7
Solve −3(x + 9)2 = −63.
Answer
(Roots) \(-9\pm\sqrt{21}\approx-13.58,-4.42\)
Solve −x2 + 4 = 2x2 − 5.
Solution
x appears twice, but they are both x2. This can be simplified if like terms are collected.
−x2 + 4 = 2x2 − 5
−3x2 + 4 = −5
Now only x2 appears, so solve by square roots.
−3x2 = −9
x2 = 3
x = \(±\sqrt{\mathbf{3}}\)
Solve x2 − 7 = 14 − 2x2
Answer
(Roots) \(\pm\sqrt7\approx\pm2.65\)
Solve x2 – 3x – 3 = 0.
Solution
Both x2 and x appear, and it already equals zero.
Try factoring.
(x – 3)(x + 1) = 0
The check step shows that this does not work.
outers + inners = middle
x + (–3x) = –2x ≠ –3x
Since factoring does not work, try using the quadratic formula which always works.
$$ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} $$
$$ x=\frac{-\left(-3\right)\pm\sqrt{\left(-3\right)^2-4\left(1\right)\left(-3\right)}}{2\left(1\right)} $$
$$ x =\frac{\mathbf{3}\pm\sqrt{\mathbf{21}}}{\mathbf{2}} $$
Solve 2x2 + 5x – 1 = 0
Answer
(Formula) \(\frac{-5\pm\sqrt{33}}{4}\approx-2.69,0.19\)
Solve x2 = 27x.
Solution
Both x2 and x appear. Set up the equation to equal zero.
x2 – 27x = 0
Try factoring. Factor the common factor first.
x(x – 27) = 0
x = 0 or x – 27 = 0
x = 0 or x = 27
Solve x2 = 81
Answer
(Roots) ±9
Find the real solutions to the equation.
Mixed Review