Use the symbols for implication ( ⇒), equivalence ( ⟺), and equality ( =).

Let a, b, c ∈ ℝ with a ≠ 0. Consider the quadratic equation ax² + bx + c = 0. (a) Calculate the discriminant Δ for the equation 2x² − 5x + 3 = 0 and use the quadratic formula to find the solutions. State whether Δ ≥ 0 ⇒ x ∈ ℝ holds in this case. (b) Calculate the discriminant Δ for the equation x² + 2x + 5 = 0 and determine whether real solutions exist. State whether x ∈ ℝ ⇒ Δ ≥ 0 holds by checking the contrapositive. (c) Based on parts (a) and (b), state using the appropriate symbol (⇒ or ⟺) the logical relationship between "Δ ≥ 0" and "the equation has real solutions".

Let a, b, c ∈ ℝ with a ≠ 0. Consider the quadratic equation ax² + bx + c = 0. (a) Prove that if Δ = b² − 4ac = 12, then the equation has two distinct real solutions. Calculate both solutions when a = 2, b = −4, c = −1. (b) Using the quadratic formula, prove that Δ ≥ 0 ⇒ the equation has real solutions. (c) Prove that the converse is also true: if the equation has real solutions, then Δ ≥ 0. (d) State, using the appropriate symbol (⇒ or ⟺), the complete logical relationship between the condition "Δ ≥ 0" and the statement "the equation has real solutions".

Consider the statement: For all real numbers $x$, if $x^2 - 5x + 6 = 0$ then $x = 2$ or $x = 3$. (a) Express this statement using the implication symbol (⇒), and verify whether the implication is true. (1 mark) (b) Determine whether the converse of this implication is also true, and hence state whether the relationship between the equation and its solutions can be expressed using the equivalence symbol (⟺). (1 mark) (c) Use your results from parts (a) and (b) to write the complete relationship between the equation $x^2 - 5x + 6 = 0$ and the statement "$x = 2$ or $x = 3$" using the appropriate symbol from {⇒, ⟺, =}. (1 mark)

Use the symbols $\Rightarrow$, $\Leftrightarrow$, or $=$ to determine the most appropriate relationship between the two statements shown in the diagram.

Let a, b, c ∈ ℝ with a ≠ 0. Consider the quadratic equation ax² + bx + c = 0. (a) Given that the discriminant Δ = b² − 4ac = 16, find the two solutions of the equation 2x² + 3x − 5 = 0 and verify that Δ ≥ 0 ⇒ x ∈ ℝ for this case. (2 marks) (b) For the general case, prove that if ax² + bx + c = 0 has real solutions, then Δ ≥ 0. (2 marks) (c) State, using the appropriate symbol (⇒ or ⟺), the logical relationship between the condition Δ ≥ 0 and the existence of real solutions to ax² + bx + c = 0. (1 mark)

Let a, b, c ∈ ℝ with a ≠ 0. Consider the quadratic equation ax² + bx + c = 0. (a) Given that the discriminant Δ = b² - 4ac = 36, calculate the two real solutions when a = 2, b = -8, and c = 1. (b) Write the relationship between Δ ≥ 0 and the existence of real solutions using the implication symbol (⇒) or the equivalence symbol (⟺). (c) Verify your answer to part (b) by showing both directions of the relationship hold.

Let a, b, c ∈ ℝ with a ≠ 0. Consider the quadratic equation ax² + bx + c = 0. (a) Given that Δ = b² − 4ac = 9, calculate the two possible values of x using the quadratic formula when a = 2, b = −3. (b) Using your answer from part (a), determine the value of c. (c) For the general case, prove that Δ ≥ 0 ⇒ x ∈ ℝ using the quadratic formula. (d) Prove the converse: that if ax² + bx + c = 0 has real solutions, then Δ ≥ 0. (e) State, using the symbol ⇒ or ⟺, the complete logical relationship between the condition "Δ ≥ 0" and the statement "ax² + bx + c = 0 has real solutions".

Let a, b, c ∈ ℝ with a ≠ 0. Consider the quadratic equation ax² + bx + c = 0. (a) Calculate the discriminant Δ for the equation 2x² - 5x + 3 = 0 and verify whether Δ ≥ 0. (b) Using the quadratic formula, calculate the solutions to 2x² - 5x + 3 = 0 and verify they are real numbers. (c) Calculate the discriminant for the equation x² + 2x + 5 = 0 and determine whether real solutions exist. (d) State, using the symbols ⇒ or ⟺, the logical relationship between the condition Δ ≥ 0 and the existence of real solutions for any quadratic equation ax² + bx + c = 0. (e) Justify your answer to part (d) by explaining both directions of the logical relationship.

Let a, b, c ∈ ℝ with a ≠ 0. Consider the quadratic equation ax² + bx + c = 0. (a) Given that Δ = b² - 4ac = 12, calculate the value of √Δ and hence evaluate the solutions x = (-b ± √Δ)/(2a) when a = 2 and b = -4. (b) For the equation 3x² - 6x + 5 = 0, calculate Δ and determine whether real solutions exist. State your conclusion using the implication symbol ⇒. (c) Prove that for any quadratic equation ax² + bx + c = 0 with real coefficients, the statement "Δ ≥ 0 if and only if the equation has real solutions" is true. State this relationship using the equivalence symbol ⟺.

Let a, b, c ∈ ℝ with a ≠ 0. Consider the quadratic equation ax² + bx + c = 0. (a) Prove that if Δ = b² − 4ac = 9, then the equation has two distinct real solutions. Calculate the solutions when a = 1, b = −1, and c = −2. (b) For the values in part (a), verify by direct substitution that x ∈ ℝ implies Δ ≥ 0. (c) State, using the symbols ⇒ or ⟺, the logical relationship between "Δ ≥ 0" and "the equation has real solutions".

Let a, b, c ∈ ℝ with a ≠ 0. Consider the quadratic equation ax² + bx + c = 0. (a) Given that a = 1, b = -4, and c = 1, calculate the discriminant Δ = b² - 4ac and hence use the quadratic formula to calculate the solutions. Verify they are real. (2 marks) (b) Given that a = 2, b = -3, and one real solution is x = 2, calculate the value of c and hence calculate Δ. (2 marks) (c) State using the symbol ⇒ or ⟺ the logical relationship between "Δ ≥ 0" and "the equation has real solutions". (1 mark)

Let a, b, c ∈ ℝ with a ≠ 0. Consider the quadratic equation ax² + bx + c = 0. (a) Prove that if Δ = b² − 4ac = 5, then the equation has two distinct real solutions. State your conclusion using the implication symbol (⇒). (b) Given that the equation has real solutions, prove that Δ ≥ 0. State your conclusion using the implication symbol (⇒). (c) Using parts (a) and (b), state the logical relationship between "Δ ≥ 0" and "the equation has real solutions" using the equivalence symbol (⟺).

Consider the quadratic equation ax² + bx + c = 0 where a, b, c ∈ ℝ and a ≠ 0. (a) Calculate the two solutions to the equation x² - 7x + 10 = 0 using the quadratic formula. Show that the discriminant Δ = b² - 4ac satisfies Δ ≥ 0, and verify that Δ ≥ 0 ⇒ the solutions are real numbers. (b) For the general quadratic equation ax² + bx + c = 0, prove that if the equation has real solutions, then Δ ≥ 0. (c) State, using the appropriate symbol (⇒ or ⟺), the complete logical relationship between the condition Δ ≥ 0 and the existence of real solutions to ax² + bx + c = 0.

Let a, b, c ∈ ℝ with a ≠ 0. Consider the quadratic equation ax² + bx + c = 0 with discriminant Δ = b² - 4ac. (a) For the quadratic equation x² - 7x + 12 = 0, calculate Δ and find the two solutions. Verify that Δ ≥ 0 ⇒ the equation has real solutions. (b) For the equation x² + 2x + 5 = 0, calculate Δ and determine whether real solutions exist. Use this to verify whether the existence of real solutions ⇒ Δ ≥ 0. (c) Based on parts (a) and (b), state using the symbols ⇒ or ⟺ the relationship between the condition Δ ≥ 0 and the existence of real solutions for a quadratic equation.

Let a, b, c ∈ ℝ with a ≠ 0. Consider the quadratic equation ax² + bx + c = 0. (a) Calculate the discriminant Δ for the equation 2x² - 8x + 3 = 0. (b) Using the quadratic formula, calculate the exact solutions to 2x² - 8x + 3 = 0. (c) For the general case ax² + bx + c = 0, prove that Δ ≥ 0 ⇒ the equation has real solutions. (d) Prove the converse: if ax² + bx + c = 0 has real solutions, then Δ ≥ 0. (e) State, using the appropriate symbol (⇒ or ⟺), the logical relationship between Δ ≥ 0 and the existence of real solutions.