Question

Solve \(2 x^2-7 x+4=0\) A. \(x=\frac{7 \pm \sqrt{17}}{4}\) B. \(x=\frac{7 \pm \sqrt{81}}{4}\) C. \(x=\frac{7 \pm 3 \sqrt{2}}{4}\) D. \(x=\frac{-4 \pm \sqrt{30}}{4}\)

Short answer

The correct solution is \( x = \frac{7 \, \pm \, \sqrt{17}}{4}\). Answer A.

Step-by-step solution

Identify the coefficients

The quadratic equation is given as \(2x^2 - 7x + 4 = 0\). Identify the coefficients: \(a = 2\), \(b = -7\), and \(c = 4\).

Apply the quadratic formula

The quadratic formula is given by: \(x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a}\). Substitute the values of \(a\), \(b\), and \(c\) into the formula.

Calculate the discriminant

Calculate the discriminant: \(b^2 - 4ac = (-7)^2 - 4 \, \cdot2\, \cdot4 = 49 - 32 = 17\).

Simplify the quadratic formula

Substitute the discriminant back into the quadratic formula: \(x = \frac{7 \, \pm \, \sqrt{17}}{4}\).

Verify the solution

Compare this solution with the options provided. The correct option is: \( x = \frac{7 \, \pm \, \sqrt{17}}{4}\).

Key concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form: \[ax^2 + bx + c = 0\]. Here’s the formula itself: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. To use the quadratic formula effectively, you need to accurately identify and plug in the values of the coefficients \(a\), \(b\), and \(c\). This formula will give you the solutions for \(x\) by calculating the expressions under the square root (the discriminant) and combining these values with the other operations in the formula. These steps are crucial for solving any quadratic equation you encounter.

Key steps:

• Identify coefficients \(a\), \(b\), and \(c\) in the quadratic equation
• Substitute these values into the quadratic formula

Discriminant

The discriminant is the part of the quadratic formula that appears under the square root symbol: \(b^2 - 4ac\). It plays a critical role in determining the nature of the roots of the quadratic equation.

The discriminant can tell you about the solutions without solving the entire equation:

• If the discriminant is positive, there are two real and distinct solutions.
• If the discriminant is zero, there is exactly one real solution (or a repeated root).
• If the discriminant is negative, there are no real solutions, but two complex ones.

For our given problem, the discriminant was calculated as follows: \(b^2 - 4ac = (-7)^2 - 4 \cdot 2 \cdot 4 = 49 - 32 = 17\). Since the discriminant is positive, this confirms that there are two real and distinct solutions for \(x\).

Coefficients

In any quadratic equation of the form \(ax^2 + bx + c = 0\), the coefficients are the numerical values that multiply the variables. The values for \(a\), \(b\), and \(c\) are crucial for applying the quadratic formula.

Let's go over them:

• \(a\): The coefficient of \(x^2\). In our equation, it is \(2\). This coefficient determines the width and direction of the parabola.
• \(b\): The coefficient of \(x\). In our equation, it is \(-7\). This coefficient influences the position of the parabola along the x-axis.
• \(c\): The constant term. In our equation, it is \(4\). This term affects the vertical position of the parabola on the y-axis.

Accurately identifying these coefficients allows for a seamless substitution into the quadratic formula, ensuring correct results.