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The quadratic formula is a powerful tool for solving quadratic equations. Quadratic equations are polynomials that have the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The quadratic formula can find the solutions (roots) of these equations using the coefficients \(a\), \(b\), and \(c\). It is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] To use this formula effectively, follow these steps:

• Identify the coefficients \(a\), \(b\), and \(c\) in the quadratic equation.
• Compute the discriminant (inside the square root) \((b^2 - 4ac)\).
• Substitute \(b\), the square root of the discriminant, and \(a\) into the quadratic formula.

The solutions will give you the x-intercepts of the function, where the graph crosses the x-axis.

The discriminant is part of the quadratic formula and significant in understanding the nature of the roots of a quadratic equation. It is represented by the expression \(b^2 - 4ac\). Here is why it matters:

• **If the discriminant is positive** \((> 0)\), the quadratic equation has two different real roots.
• **If the discriminant is zero** \((= 0)\), the quadratic equation has exactly one real root; the parabola touches the x-axis at one point.
• **If the discriminant is negative** \((< 0)\), the quadratic equation has two complex roots; the parabola does not intersect the x-axis.

In our example, the discriminant is 5. Since 5 is positive, the quadratic equation \(x^2 + 5x + 5 = 0\) has two different real solutions.

Solving quadratic equations is vital in finding the points where the function equals zero. Let’s review the process step-by-step using the given quadratic function \(y = x^2 + 5x + 5\).
**Step 1: Set the equation to zero**
To find the x-intercepts, we set the function equal to zero. This gives us: \[ x^2 + 5x + 5 = 0 \]
**Step 2: Identify a, b, and c**
In the equation \(x^2 + 5x + 5 = 0\), the coefficients are \(a = 1\), \(b = 5\), and \(c = 5\).
**Step 3: Compute the discriminant**
The discriminant is calculated using \( \Delta = b^2 - 4ac \). Substituting the values, we get: \[ \Delta = 5^2 - 4(1)(5) = 25 - 20 = 5 \]
**Step 4: Use the quadratic formula**
Plug the values of \(a\), \(b\), and the discriminant back into the quadratic formula: \[ x = \frac{-5 \pm \sqrt{5}}{2} \] Solving this gives the two x-intercepts. This technique ensures that you can find the solutions to any quadratic equation, given you know the coefficients.