Solving Quadratic Equations Solve the quadratic equation \( -5 x^{2}-50 x=45 \) by using the Quadratic Formula. Verify your result by graphing and using the Intersect Method. Steps 1. Write the equation in Standard Form 2. Identify the coefficients \( a, b \) and \( c \). 3. Substitute the values into the quadratic equation \[ x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} \] 4. Solve the equation for \( x_{1} \) and \( x_{2} \) \[ x_{1}=\frac{-b-\sqrt{b^{2}-4 a c}}{2 a} \quad x_{2}=\frac{-b+\sqrt{b^{2}-4 a c}}{2 a} \] 5. Write your answers in Exact Form and in Approximate Form (Rounded to three decimal places as needed). Note that in some cases, the Exact Form and the Approximate Form may be the same. 6. Note: If only one solution exists, \( x_{2} \) will equal DNE \[ -5 x^{2}-50 x=45 \] \[ x_{1}=\square \text { Exact Form } \] Approximate Form \[ x_{1}= \] \( \square \) \[ \begin{array}{l} x_{2}=\square \text { Exact Form } \\ x_{2}=\square \text { Approximate Form } \end{array} \]

First, let's rewrite the equation in standard form. To do that, we need to move all terms to one side, which gives us: \[ -5x^2 - 50x - 45 = 0 \] Now we can identify the coefficients: - \( a = -5 \) - \( b = -50 \) - \( c = -45 \) Next, we’ll substitute these values into the quadratic formula: \[ x = \frac{-(-50) \pm \sqrt{(-50)^{2} - 4 \cdot (-5) \cdot (-45)}}{2 \cdot (-5)} \] Calculating the discriminant: \[ (-50)^{2} - 4 \cdot (-5) \cdot (-45) = 2500 - 900 = 1600 \] Now let's continue substituting: \[ x = \frac{50 \pm \sqrt{1600}}{-10} \] \[ \sqrt{1600} = 40 \] \[ x_{1} = \frac{50 - 40}{-10} = \frac{10}{-10} = -1 \quad \text{Exact Form} \] \[ x_{2} = \frac{50 + 40}{-10} = \frac{90}{-10} = -9 \quad \text{Exact Form} \] In Approximate Form (though these are whole numbers): \[ x_{1} = -1.000 \quad x_{2} = -9.000 \] Just to keep it light, did you know the origins of the quadratic formula are traced back to ancient civilizations? The Babylonians had methods for solving quadratic equations as far back as 2000 BC, relying on geometry! It’s fascinating how math has evolved through the ages, transforming methods into the elegant formulas we use today. Speaking of transformation, real-world applications of quadratic equations are abundant! From physics problems involving projectile motion to calculating areas in architecture, quadratics help us model and predict various phenomena around us. Embracing these formulas can turn abstract concepts into practical tools, making quadratic equations not just something we solve in classrooms, but skills we can use out in the world!