Question

1. look at this quadratic equation and explain what you think will be the best approach to solving it. do not solve the equation.

3.38x² + 2.72x - 9.31 = 0

1. solve the quadratic equation by any means. identify the method and explain why you chose it. irrational answers may be left in radical form or approximated with a calculator (round to two decimal places).

36x² - 64 = 0

1. solve the quadratic equation by any means. identify the method and explain why you chose it. irrational answers may be left in radical form or approximated with a calculator (round to two decimal places).

4x² - 4x - 3 = 2

Explanation:

1. Explanation for the first equation \(3.38x^{2}+2.72x - 9.31=0\):

• The quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) (where for the general quadratic equation \(ax^{2}+bx + c = 0\)) is the best approach.
• Reason: The coefficients \(a = 3.38\), \(b=2.72\), and \(c=-9.31\) are non - integer and non - simple fractions. Factoring would be very difficult (if possible at all) due to the decimal values. Completing the square would also be cumbersome with these decimal coefficients.

1. Solving the second equation \(36x^{2}-64 = 0\):

• Method: Using the difference of squares formula \(a^{2}-b^{2}=(a + b)(a - b)\) (or simply solving for \(x^{2}\) and then taking square roots).
• Steps:
• First, rewrite the equation as \(36x^{2}=64\).
• Then \(x^{2}=\frac{64}{36}=\frac{16}{9}\).
• Taking square roots, \(x=\pm\sqrt{\frac{16}{9}}=\pm\frac{4}{3}\approx\pm1.33\).
• Reason for the method: The equation is of the form \(ax^{2}-c = 0\) (a special case of a quadratic equation \(ax^{2}+bx + c = 0\) where \(b = 0\)). It can be easily solved by isolating \(x^{2}\) and then taking square roots. Also, \(36x^{2}-64=(6x)^{2}-8^{2}\), which is a difference of squares.

1. Solving the third equation \(4x^{2}-4x - 3=2\) (rewrite as \(4x^{2}-4x - 5 = 0\)):

• Method: Quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) (where \(a = 4\), \(b=-4\), \(c=-5\)).
• Steps:
• Calculate the discriminant \(\Delta=b^{2}-4ac=(-4)^{2}-4\times4\times(-5)=16 + 80=96\).
• Then \(x=\frac{4\pm\sqrt{96}}{8}=\frac{4\pm4\sqrt{6}}{8}=\frac{1\pm\sqrt{6}}{2}\).
• Approximating, \(x=\frac{1+\sqrt{6}}{2}\approx\frac{1 + 2.45}{2}=1.73\) and \(x=\frac{1-\sqrt{6}}{2}\approx\frac{1-2.45}{2}=-0.73\).
• Reason for the method: Factoring is not straightforward for the quadratic \(4x^{2}-4x - 5\). Completing the square would involve more steps compared to using the quadratic formula which is a direct formula for solving any quadratic equation \(ax^{2}+bx + c = 0\).

Answer:

1. Quadratic formula is the best approach.
2. \(x=\pm\frac{4}{3}\approx\pm1.33\) (using difference of squares or square - root method).
3. \(x\approx1.73\) and \(x\approx - 0.73\) (using quadratic formula).