Answer:
[tex]w_1=2\\w_2=-9[/tex]
Step-by-step explanation:
This is going to be solved using the quadratic formula.
1. Determine the quadratic equation’s coefficient [tex]a[/tex], [tex]b[/tex], and [tex]c[/tex].
Use the standard form, [tex]ax^2+bx+c=0[/tex] , to find the coefficients of our equation,
[tex]w^2+7w-18=0[/tex]:
a = 1
b = 7
c = -18
2. Plug these coefficients into the quadratic formula
The quadratic formula gives us the roots for [tex]aw^2+bw+c=0[/tex] , in which [tex]a[/tex], [tex]b[/tex], and [tex]c[/tex] are numbers (or coefficients), as follows:
[tex]w=\frac{-b\pm\sqrt{b^2-4ac} }{2a}[/tex]
7 additional steps:
a = 1
b = 7
c = -18
[tex]w=\frac{-7\pm\sqrt{7^2=4*1*-18} }{2*1}[/tex]
Simplify exponents and square roots
[tex]w=\frac{-7\pm\sqrt{49-4*1*-18} }{2*1}[/tex]
Perform any multiplication or division, from left to right:
[tex]w=\frac{-7\pm\sqrt{49-4*-18}}{2*1}\\w=\frac{-7\pm\sqrt{49--72}}{2*1}\\\\w=\frac{-7\pm\sqrt{49+72}}{2*1}\\w=\frac{-7\pm\sqrt{121}}{2*1}[/tex]
to get the result:
[tex]w=\frac{-7\pm\sqrt{49-4*-18}}{2}[/tex]
3. Simplify square root [tex]\sqrt{121}[/tex]
Simplify 121 by finding its prime factors:
121 ( 11 * 11)
The prime factorization of 121 is 11^2
Write the prime factors:
[tex]\sqrt{121}=\sqrt{11\cdot 11}[/tex]
Group the prime factors into pairs and rewrite them in exponent form:
'[tex]\sqrt{11\cdot 11}=\sqrt{11^2}[/tex]
Use the rule [tex]\sqrt{x^2=x}[/tex] to simplify further:
[tex]\sqrt{11^2}=11[/tex]
4. Solve the equation for w
[tex]w=\frac{-7\pm11}{2}[/tex]
The ± means two answers are possible.
Separate the equations:
[tex]w_1=\frac{(-7+11)}{2}[/tex] and [tex]w_2=\frac{-7-11}{2}[/tex]
[tex]w_1=\frac{-7+11}{2}[/tex]
[tex]w_1=\frac{4}{2}[/tex]
[tex]w_1=2[/tex]
[tex]w_2=\frac{-7-11}{2}[/tex]
[tex]w_2=\frac{-18}{2}[/tex]
[tex]w_2=-9[/tex]
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Why learn this
- In their most basic function, quadratic equations define shapes like circles, ellipses and parabolas. These shapes can in turn be used to predict the curve of an object in motion, such as a ball kicked by football player or shot out of a cannon. When it comes to an object’s movement through space, what better place to start than space itself—with the revolution of planets around the sun in our solar system. The quadratic equation was used to establish that planets’ orbits are elliptical, not circular. Determining the path and speed an object travels through space is possible even after it has come to a stop: the quadratic equation can calculate how fast a vehicle was moving when it crashed. With information like this, the automotive industry can design brakes to prevent collisions in the future. Many industries use the quadratic equation to predict and thus improve their products’ lifespan and safety.
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Terms and topics
- Solving quadratic equations using the quadratic formula
- Simplifying radicals
- Find prime factors
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