Which equation shows the quadratic formula used correctly to solve \( 7x^2 = 9 + x \) for \( x \)?
December 15, 2024
Question: Which equation shows the quadratic formula used correctly to solve 7x2=9+x for x?
Answer: . x=2(7)−1±12−4(7)(9)
Explanation:
Step 1: Rearranging the equation into standard quadratic form
The given equation is 7x2=9+x. To use the quadratic formula, we first need to rewrite it in the form ax2+bx+c=0.
7x2−x−9=0
Step 2: Identifying the coefficients
Now the quadratic equation is in the form ax2+bx+c=0, where:
a=7
b=−1
c=−9
Step 3: Applying the quadratic formula
The quadratic formula is given by:
x=2a−b±b2−4ac
Substituting the values of a, b, and c into the formula:
x=2(7)−(−1)±(−1)2−4(7)(−9)
This simplifies to:
x=141±1+252x=141±253
Thus, the correct form of the equation is:
x=2(7)−1±12−4(7)(9)
Extended Knowledge:
Quadratic Formula
The quadratic formula is a method for solving any quadratic equation of the form ax2+bx+c=0. It is derived from completing the square of the equation and provides the solutions for x. The two possible solutions are given by the ± symbol, indicating two possible values for x.
Completing the Square
Completing the square is a method used to solve quadratic equations by rewriting the equation in a perfect square form. This method ultimately leads to the quadratic formula and can be an alternative way to find the roots of a quadratic equation.
Similar Questions
Question 1: Solve the quadratic equation 3x2−6x+2=0 using the quadratic formula.
Answer:
x=2(3)−(−6)±(−6)2−4(3)(2)
Explanation:
Step 1: Identify the coefficients
For the equation 3x2−6x+2=0, we have:
a=3
b=−6
c=2
Step 2: Apply the quadratic formula
The quadratic formula is:
x=2a−b±b2−4ac
Substituting the values of a, b, and c into the formula:
x=2(3)−(−6)±(−6)2−4(3)(2)=66±36−24=66±12
Step 3: Simplify the result
x=66±23=1±33
Thus, the solutions for x are:
x=1+33,x=1−33
Extended Knowledge:
Quadratic Formula
The quadratic formula x=2a−b±b2−4ac is used to solve any quadratic equation of the form ax2+bx+c=0. It can provide both real and complex roots depending on the discriminant (b2−4ac).
Question 2: Solve the quadratic equation 4y2+8y−5=0 using the quadratic formula.
Answer:
y=2(4)−8±82−4(4)(−5)
Explanation:
Step 1: Identify the coefficients
For the equation 4y2+8y−5=0, we have:
a=4
b=8
c=−5
Step 2: Apply the quadratic formula
y=2(4)−8±82−4(4)(−5)=8−8±64+80=8−8±144
Step 3: Simplify the result
y=8−8±12
Thus, the solutions for y are:
y=8−8+12=84=21,y=8−8−12=8−20=−25
Question 3: Solve the quadratic equation 2z2−3z+1=0 using the quadratic formula.
Answer:
z=2(2)−(−3)±(−3)2−4(2)(1)
Explanation:
Step 1: Identify the coefficients
For the equation 2z2−3z+1=0, we have:
a=2
b=−3
c=1
Step 2: Apply the quadratic formula
z=2(2)−(−3)±(−3)2−4(2)(1)=43±9−8=43±1
Step 3: Simplify the result
z=43±1
Thus, the solutions for z are:
z=43+1=44=1,z=43−1=42=21
Question 4: Solve the quadratic equation 5t2+7t−3=0 using the quadratic formula.