Problem 30
Question
\(d^{2}-6 d-16=0\)
Step-by-Step Solution
Verified Answer
The solutions are d_1 = 8 and d_2 = -2.
1Step 1: Identify the equation type
The given equation is a quadratic equation in the form of \(ax^2 + bx + c = 0\). Here, determine that it is in the form a = 1, b = -6, and c = -16.
2Step 2: Use the quadratic formula
Write down the quadratic formula: \[d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. Substitute the values of a, b, and c into the formula.
3Step 3: Calculate the discriminant
Calculate the discriminant \(b^2 - 4ac\). For the given values, \((-6)^2 - 4\cdot1\cdot(-16) = 36 + 64 = 100\). The discriminant is 100.
4Step 4: Solve for d
Using the quadratic formula, substitute the discriminant back in: \[d = \frac{-(-6) \pm \sqrt{100}}{2\cdot1}\]. This simplifies to: \[d = \frac{6 \pm 10}{2}\].
5Step 5: Find the roots
Calculate the two possible values for d: \[d_1 = \frac{6 + 10}{2} = 8\] and \[d_2 = \frac{6 - 10}{2} = -2\]. Thus, the roots are d_1 = 8 and d_2 = -2.
Key Concepts
Quadratic FormulaDiscriminantQuadratic Equation Roots
Quadratic Formula
To solve a quadratic equation, one powerful tool we use is the quadratic formula. The quadratic formula helps you find the values of the variable that make the equation true. This is especially useful when factoring is difficult or impossible.
The quadratic formula is: d = \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
The letters a, b, and c are coefficients from the quadratic equation in the form ax² + bx + c = 0.
To apply the quadratic formula, plug in the values of a, b, and c into the formula. This method will give you the possible values for the variable that satisfies the quadratic equation.
The quadratic formula is: d = \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
The letters a, b, and c are coefficients from the quadratic equation in the form ax² + bx + c = 0.
- a is the coefficient of x²
- b is the coefficient of x
- c is the constant term
To apply the quadratic formula, plug in the values of a, b, and c into the formula. This method will give you the possible values for the variable that satisfies the quadratic equation.
Discriminant
A key part of the quadratic formula is the discriminant, located under the square root symbol: \(b^2 - 4ac\). The discriminant tells us about the nature of the roots of the quadratic equation.
In our exercise, the discriminant calculation for the equation \(d^2 - 6d - 16 = 0\) is: \( (-6)^2 - 4 \cdot 1 \cdot (-16) = 100\). Since 100 is positive, this tells us there are two distinct real roots.
- If the discriminant is positive, there are two real and distinct roots.
- If the discriminant is zero, there is one real root, sometimes called a 'duplicate' root.
- If the discriminant is negative, there are no real roots but two complex roots.
In our exercise, the discriminant calculation for the equation \(d^2 - 6d - 16 = 0\) is: \( (-6)^2 - 4 \cdot 1 \cdot (-16) = 100\). Since 100 is positive, this tells us there are two distinct real roots.
Quadratic Equation Roots
The roots of a quadratic equation are the solutions to the equation; they are where the graph intersects the x-axis. Once the discriminant value is calculated, it is substituted back into the quadratic formula to find these roots.
From our example, with the discriminant of 100: \(\frac{-(-6) \pm \sqrt{100}}{2 \cdot 1}\), simplifies to \(\frac{6 \pm 10}{2}\). This results in two calculations:
The solutions or roots of the equation \(d^2 - 6d - 16 = 0\) are d = 8 and d = -2. These roots show the points where the parabola crosses the x-axis, representing the solutions to our quadratic equation.
From our example, with the discriminant of 100: \(\frac{-(-6) \pm \sqrt{100}}{2 \cdot 1}\), simplifies to \(\frac{6 \pm 10}{2}\). This results in two calculations:
- \(\frac{6 + 10}{2} = 8\)
- \(\frac{6 - 10}{2} = -2\)
The solutions or roots of the equation \(d^2 - 6d - 16 = 0\) are d = 8 and d = -2. These roots show the points where the parabola crosses the x-axis, representing the solutions to our quadratic equation.