Problem 9 Use the quadratic formula to sol... [FREE SOLUTION] (2024)

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Quadratic equations are mathematical expressions of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( x \) represents the variable. To solve these equations, one effective method is the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]?. This formula helps find the values of \( x \) that satisfy the equation.

First, ensure the equation is in its standard form. For example, given \( 2x^2 - 2x = 1 \), transform it into \( 2x^2 - 2x - 1 = 0 \). Next, identify the coefficients, plug them into the formula, and simplify each step.

Identifying the coefficients \( a \), \( b \), and \( c \) is essential in solving a quadratic equation using the quadratic formula.

For instance, in the equation \( 2x^2 - 2x - 1 = 0\):

• \( a \) is the coefficient of \( x^2 \), so \( a = 2 \).
• \( b \) is the coefficient of \( x \), so \( b = -2 \).
• \( c \) is the constant term, so \( c = -1 \).

After identifying these coefficients, you can substitute them in the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the solution.

Simplifying expressions is an important step in solving the quadratic equation. Once you have substituted the coefficients into the quadratic formula, you need to simplify each part step by step.

Start by working under the square root (the discriminant):\[ b^2 - 4ac \]. For our example with \( b = -2 \), \( a = 2 \), and \( c = -1 \), it becomes: \( (-2)^2 - 4 \cdot 2 \cdot (-1) = 4 + 8 = 12 \).

Next, simplify the square root: \( \sqrt{12} \) simplifies to \( 2\sqrt{3} \). Finally, simplify the fraction step by step to get the final solution:

• Simplify the numerator: \( 2 \pm 2\sqrt{3} \)
• Factor out common terms if possible and simplify the fraction: \( \frac{2(1 \pm \sqrt{3})}{4} = \frac{1 \pm \sqrt{3}}{2} \)

Square roots are crucial in the quadratic formula as they often determine the solutions' complexity. In the quadratic formula, you will encounter the term \( \sqrt{b^2 - 4ac} \), where \( b^2 - 4ac \) is the discriminant.

Simplifying square roots is essential. For instance, \( \sqrt{12} \) simplifies to \( 2\sqrt{3} \). This is because 12 can be factored into \( 4 \times 3 \), and since \( \sqrt{4} \) is 2, it evolves into \( 2\sqrt{3} \).

Simplifying square roots helps simplify the quadratic formula and obtain a clean final solution. The expressions become easier to handle, making it simpler to interpret the final solutions: \[ x = \frac{1 + \sqrt{3}}{2} \] and \[ x = \frac{1 - \sqrt{3}}{2} \].