Solve $x^2+4 x-7=0$
Answer (2)
Recognize the problem as solving a quadratic equation.
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Calculate the discriminant: D = b 2 − 4 a c = 44 .
Find the roots: x = − 2 ± 11 . The solutions are x 1 = − 2 − 11 and x 2 = − 2 + 11 .
Explanation
Problem Analysis The problem has two parts:
Express complex numbers in terms of i .
Solve the quadratic equation x 2 + 4 x − 7 = 0 .
Solution Plan Part 1: Any complex number can be written in the form a + bi , where a and b are real numbers and i is the imaginary unit, defined as i = − 1 . This part of the question is more of a definition/concept recall.
Part 2: We need to solve the quadratic equation x 2 + 4 x − 7 = 0 . We can use the quadratic formula to find the solutions.
Quadratic Formula The quadratic formula is given by x = 2 a − b ± b 2 − 4 a c where a = 1 , b = 4 , and c = − 7 in our equation x 2 + 4 x − 7 = 0 .
Calculating the Discriminant First, calculate the discriminant ( D ): D = b 2 − 4 a c = 4 2 − 4 ( 1 ) ( − 7 ) = 16 + 28 = 44 Since the discriminant is positive, we have two distinct real roots.
Applying the Formula and Finding the Roots Now, apply the quadratic formula: x = 2 ( 1 ) − 4 ± 44 = 2 − 4 ± 2 11 = − 2 ± 11 Thus, the two solutions are x 1 = − 2 − 11 and x 2 = − 2 + 11 .
Final Answer The solutions to the quadratic equation x 2 + 4 x − 7 = 0 are x = − 2 − 11 and x = − 2 + 11 .
Examples
Quadratic equations are used in various real-world applications, such as calculating the trajectory of a projectile, designing parabolic mirrors and antennas, and determining the dimensions of a rectangular area given its perimeter and area. For instance, if you're launching a rocket, you can use a quadratic equation to model its path and predict where it will land, taking into account factors like initial velocity and gravity. Similarly, architects and engineers use quadratic equations to design structures and ensure their stability and optimal performance.
To solve the equation x 2 + 4 x − 7 = 0 , we used the quadratic formula and calculated the discriminant. The solutions are x = − 2 − 11 and x = − 2 + 11 . This indicates the x-values where the quadratic crosses the x-axis.
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