Math Problem Statement

Complete the table for y = x^2 - 5x + 6, draw the graph, and find the solutions for x^2 - 5x = x - 7 by drawing a suitable line on the graph.

Solution

Let's analyze the questions from the image:

Part (a) - Completing the table for y=x2−5x+6y = x^2 - 5x + 6

The equation provided is y=x2−5x+6y = x^2 - 5x + 6, and you are asked to complete the table for different values of xx. Let's substitute each value of xx into the equation:

  • For x=0x = 0: y=02−5(0)+6=6y = 0^2 - 5(0) + 6 = 6
  • For x=1x = 1: y=12−5(1)+6=1−5+6=2y = 1^2 - 5(1) + 6 = 1 - 5 + 6 = 2
  • For x=2x = 2: y=22−5(2)+6=4−10+6=0y = 2^2 - 5(2) + 6 = 4 - 10 + 6 = 0
  • For x=3x = 3: y=32−5(3)+6=9−15+6=0y = 3^2 - 5(3) + 6 = 9 - 15 + 6 = 0
  • For x=4x = 4: y=42−5(4)+6=16−20+6=2y = 4^2 - 5(4) + 6 = 16 - 20 + 6 = 2
  • For x=5x = 5: y=52−5(5)+6=25−25+6=6y = 5^2 - 5(5) + 6 = 25 - 25 + 6 = 6

So, the complete table is:

x & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline y & 6 & 2 & 0 & 0 & 2 & 6 \\ \end{array}$$ ### Part (b) - Drawing the graph of $$y = x^2 - 5x + 6$$ The graph shown depicts a parabola, which corresponds to the equation $$y = x^2 - 5x + 6$$. The points from the table that should be plotted are: - $$(0, 6)$$ - $$(1, 2)$$ - $$(2, 0)$$ - $$(3, 0)$$ - $$(4, 2)$$ - $$(5, 6)$$ This confirms the parabolic shape you've drawn, and it's symmetric around $$x = 2.5$$, which is the axis of symmetry. ### Part (c) - Finding solutions of $$x^2 - 5x = x - 7$$ To solve $$x^2 - 5x = x - 7$$, first rearrange it as: $$x^2 - 5x - x + 7 = 0 \implies x^2 - 6x + 7 = 0$$ Now, this is the quadratic equation you need to solve. By graphing, you can plot the line $$y = x - 7$$ on the same graph as $$y = x^2 - 5x + 6$$ and find the points where these two curves intersect. These points are the approximate solutions to the equation. The solutions would be the x-values where the line crosses the parabola. Let me know if you would like further assistance in plotting the line or solving this equation algebraically. --- Do you have any questions or need additional details? Here are 5 related questions to explore further: 1. Can you explain why the graph of a quadratic function is always a parabola? 2. How can we derive the axis of symmetry for any quadratic equation? 3. Why do we rearrange equations like in part (c) before solving them graphically? 4. What other methods can we use to solve quadratic equations besides graphing? 5. Can you estimate the solutions of $$x^2 - 6x + 7 = 0$$ from the graph? **Tip:** When solving a quadratic equation, always consider both graphing and algebraic methods, as each can provide unique insights into the nature of the solutions.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Graphing
Equations

Formulas

y = ax^2 + bx + c for a quadratic function
Standard form of a quadratic equation: ax^2 + bx + c = 0

Theorems

Quadratic formula
Properties of parabolas

Suitable Grade Level

Grades 8-10

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