Question

Which equation has one real solution? Explain. (A) \(3 x^2+4=-2\left(x^2+8\right)\) (B) \(5 x^2-4=x^2-4\) (C) \(2(x+3)^2=18\) (D) \(\frac{3}{2} x^2-5=19\)

Short answer

(B) \(5 x^2-4=x^2-4\) is the equation with one real solution.

Step-by-step solution

Standardize and find discriminant for Option A

Rearrange equation (A) to standardize: \(3x^2+2x^2-4=0 \rightarrow 5x^2-4=0\). Calculate the discriminant: \(D=b^{2}-4ac=0^{2}-4*5*(-4)=80\). Because \(D>0\), equation (A) will have two real solutions.

Standardize and find discriminant for Option B

Rearrange equation (B) to standardize: \(5x^2-x^2=0 \rightarrow 4x^2=0\). Calculate the discriminant: \(D=b^{2}-4ac=0^{2}-4*4*0=0\). Because \(D=0\), equation (B) will have one real solution.

Standardize and find discriminant for Option C

Rearrange equation (C) to standardize: \(2x^2+12x+18-18=0 \rightarrow 2x^2+12x=0\). Calculate the discriminant: \(D=b^{2}-4ac=12^{2}-4*2*0=144\). Because \(D>0\), equation (C) will have two real solutions.

Standardize and find discriminant for Option D

Rearrange equation (D) to standardize: \(\frac{3}{2}x^2-24=0\). Calculate the discriminant: \(D=b^{2}-4ac=0^{2}-4*\frac{3}{2}*(-24)=144\). Because \(D>0\), equation (D) will have two real solutions.

Key concepts

Quadratic Formula

The quadratic formula is a keystone in solving quadratic equations. It is derived from the process of completing the square, and it offers a straightforward method for finding the roots of any quadratic equation. The formula is expressed as:
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]
Where \(a\), \(b\), and \(c\) represent the coefficients of the equation in its standard form, which is \(ax^2 + bx + c = 0\). The term under the square root sign, \(b^2 - 4ac\), is known as the discriminant. The quadratic formula can solve all types of quadratic equations, including those with real and complex solutions, by adjusting the discriminant value accordingly.

Real Solutions of Quadratics

Understanding the nature of real solutions for quadratic equations is crucial for students. The discriminant of the quadratic equation, denoted as \(D\) and given by the formula \(D=b^2-4ac\), determines the number and type of solutions. There are three possible scenarios based on the value of the discriminant:

• If \(D>0\), the quadratic equation has two distinct real solutions.
• If \(D=0\), there is exactly one real solution, also known as a repeated or double root.
• If \(D<0\), there are no real solutions; instead, there are two complex solutions.

For example, in the exercise provided, option B (\(5x^2 - 4 = x^2 - 4\)) will have one real solution because the discriminant is zero. This condition satisfies the requirement for an equation to have a unique solution in the real number system.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is an essential concept and is written as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a\) is not zero. This specific arrangement allows for the use of the quadratic formula and the discriminant to find the roots of the equation. Converting an equation to this form, as demonstrated in the steps to solve the original exercise, often involves combining like terms and simplifying the equation.
For instance, to solve for option C in the exercise, we start by expanding and rearranging to get \(2x^2 + 12x = 0\), which is in the standard form with \(a = 2\), \(b = 12\), and \(c = 0\). From this form, one can easily calculate the discriminant and determine the number of real solutions.