Quadratic Inequalities Quiz

Solving the Inequality

• The first step is to rearrange the terms to get all the terms on one side of the inequality, making it a quadratic expression.
• The inequality can be rewritten as $x^2 - x - 7 > 0$ by subtracting $x+3$ from both sides.
• To find the solutions of the quadratic inequality, the quadratic formula can be applied.
• The quadratic formula provides the roots of the equation $ax^2+bx+c=0$, where a, b, and c are coefficients.
• In this case, a = 1, b = -1, and c = -7.

Finding the Solutions

• The solutions to the inequality $x^2 - x - 7 > 0$ are the values of x that make the expression $x^2 - x - 7$ greater than zero.
• The solutions to $x^2 - x - 7 = 0$ are the values of x where the graph of the function $y = x^2 - x - 7$ intersects the x-axis.
• Solving the equation using the quadratic formula gives the solutions:
• $x = (1 \pm \sqrt{1 + 28}) / 2$
• $x = (1 \pm \sqrt{29}) / 2$
• These are the x-intercepts of the function $y = x^2 - x - 7$.
• Because the inequality is greater than zero, the solutions will be the intervals of the x-axis where the graph is above the x-axis.
• These intervals are $x < (1 - sqrt{29})/2$ or $x > (1 + sqrt{29})/2$.

Graphing the Function

• The graph of the function $y = x^2 - x - 7$ is a parabola that opens upwards.
• The parabola intersects the x-axis at two points which correspond to the solutions of the quadratic equation: $(1 - sqrt{29}) / 2$ and $(1 + sqrt{29}) / 2$.
• The x-intercepts are where the function equals zero.
• The graph of the function $y = x^2 - x - 7$ is a parabola that opens upwards and passes through the x-axis at the two solutions.
• The solution intervals are the areas where the parabola is above the x-axis, meaning y is greater than zero.