olve for the roots in simplest form using the quadratic formula:4, x, squared, minus, 28, x, equals, minus, 374x 2 −28x=−37

1. Break Down the Problem

We start with the equation: $4x^2 - 28x = -37$ We will bring all terms to one side of the equation to rewrite it in standard form.

2. Relevant Concepts

The standard form of a quadratic equation is: $ax^2 + bx + c = 0$ In our case, we need to identify $a$, $b$, and $c$ for the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

3. Analysis and Detail

Step 1: Rearranging the equation: $4x^2 - 28x + 37 = 0$ Here, $a = 4$, $b = -28$, and $c = 37$.

Step 2: Calculating the discriminant ($D$): $D = b^2 - 4ac = (-28)^2 - 4 \cdot 4 \cdot 37$ Calculating $D$: $D = 784 - 592 = 192$

Step 3: Applying the quadratic formula: $x = \frac{-(-28) \pm \sqrt{192}}{2 \cdot 4}$ This simplifies to: $x = \frac{28 \pm \sqrt{192}}{8}$

Step 4: Simplifying $\sqrt{192}$: $\sqrt{192} = \sqrt{64 \cdot 3} = 8\sqrt{3}$ Thus, the equation for $x$ becomes: $x = \frac{28 \pm 8\sqrt{3}}{8}$ This can be simplified further: $x = \frac{28}{8} \pm \frac{8\sqrt{3}}{8}$ So: $x = 3.5 \pm \sqrt{3}$

4. Verify and Summarize

The roots of the quadratic equation are: $x = 3.5 + \sqrt{3} \quad \text{and} \quad x = 3.5 - \sqrt{3}$

Final Answer

The roots in simplest form are: $x = 3.5 + \sqrt{3} \quad \text{and} \quad x = 3.5 - \sqrt{3}$