Examkin

Soru

Zorluk: OrtaQuadratic Equations

In the quadratic equation 3x2−15x+c=03x^2 - 15x + c = 0, cc is a constant. If one of the solutions to the equation is 22, what is the value of the other solution?

  1. 3Cevap
  2. B
    -3
  3. C
    -7
  4. D
    5

Cevap

The other solution to the equation is 33.
Substituting x=2x = 2 into the quadratic equation gives 3(2)2−15(2)+c=03(2)^2 - 15(2) + c = 0, which simplifies to 12−30+c=012 - 30 + c = 0, or c=18c = 18. Substituting c=18c = 18 back into the original equation yields 3x2−15x+18=03x^2 - 15x + 18 = 0. Dividing the entire equation by 33 results in x2−5x+6=0x^2 - 5x + 6 = 0. Factoring this equation gives (x−2)(x−3)=0(x - 2)(x - 3) = 0, which means the solutions are x=2x = 2 and x=3x = 3. Therefore, the other solution is 33. Alternatively, the sum of the roots of a quadratic equation in the form ax2+bx+c=0ax^2 + bx + c = 0 is given by −b/a-b/a. For this equation, the sum of the roots is −(−15)/3=5-(-15)/3 = 5. Since one root is 22, the other root must be 5−2=35 - 2 = 3.

Adım Adım Çözüm

1
Substitute the given solution x=2x = 2 into the quadratic equation to find the value of the constant cc.
3(2)2−15(2)+c=0  ⟹  12−30+c=0  ⟹  c=183(2)^2 - 15(2) + c = 0 \implies 12 - 30 + c = 0 \implies c = 18
Since 22 is a solution, it must satisfy the equation when substituted for xx.
2
Substitute c=18c = 18 back into the original quadratic equation and simplify by dividing both sides by 33.
3x2−15x+18=0  ⟹  x2−5x+6=03x^2 - 15x + 18 = 0 \implies x^2 - 5x + 6 = 0
Dividing the equation by the greatest common factor simplifies the expression and makes it easier to factor.
3
Factor the simplified quadratic equation to find both solutions.
(x−2)(x−3)=0  ⟹  x=2(x - 2)(x - 3) = 0 \implies x = 2 or x=3x = 3
The solutions to the factored equation are the values of xx that make each factor equal to zero.

Anahtar Kavram

Solving quadratic equations by substitution and factoring
Tahmini Süre:1m 30s
Bu soruyu puanla

Konular

Quadratic Equations

Bu konudan daha fazla soru